A novel theoretical framework for simultaneous measurement of excitatory and inhibitory conductances.

The firing of neurons throughout the brain is determined by the precise relations between excitatory and inhibitory inputs, and disruption of their balance underlies many psychiatric diseases. Whether or not these inputs covary over time or between repeated stimuli remains unclear due to the lack of experimental methods for measuring both inputs simultaneously. We developed a new analytical framework for instantaneous and simultaneous measurements of both the excitatory and inhibitory neuronal inputs during a single trial under current clamp recording. This can be achieved by injecting a current composed of two high frequency sinusoidal components followed by analytical extraction of the conductances. We demonstrate the ability of this method to measure both inputs in a single trial under realistic recording constraints and from morphologically realistic CA1 pyramidal model cells. Future experimental implementation of our new method will facilitate the understanding of fundamental questions about the health and disease of the nervous system.

Introduction

Neuronal firing is orchestrated by the interplay of excitatory and inhibitory inputs. Therefore, studying their relationship has been crucial to solving fundamental questions in cellular and system neuroscience. Disrupted relations between these inputs were suggested to accompany many neurological diseases and in particular epileptic seizures. It is commonly believed that such seizures are accompanied and even caused by a disruption of excitation-inhibition ratio and their temporal relationships [1-3].

The most widely used method to measure inhibitory and excitatory inputs in isolation is the voltage clamp technique. To reveal excitatory synaptic currents the membrane potential is voltage clamped near the reversal potential of inhibition (near -80 mV) and inhibitory synaptic currents are revealed when the voltage is clamped near the excitatory reversal potential (near 0 mV). Voltage clamp recordings have been used in this manner to study mechanisms of feature selectivity of cortical cells belonging to various modalities [4-13]. Current clamp recordings also allow for the isolation of excitatory and inhibitory conductances, which is done by injecting constant positive or negative currents which bring the membrane potential near the reversal potential of these two input types [8-10,14-18].

Voltage and current clamp approaches share several similarities. In both cases, excitation and inhibition are recorded in different trials and conductances are estimated by fitting the averaged data with the membrane potential equation (Eq 1 below). Hence, these methods provide only an average picture and thus fail to capture the instantaneous and trial-by-trial based insight into the relations between excitation and inhibition.

The instantaneous relation between excitation and inhibition in-vivo was revealed using a different approach, relying on the finding that the membrane potential of nearby cortical cells in anesthetized animals is highly synchronized [19,20]. This approach consists of depolarizing one cell to reveal its inhibitory inputs while simultaneously hyperpolarizing a neighboring cell to reveal its excitatory inputs. Doing this showed that excitatory and inhibitory synaptic inputs are highly correlated in anesthetized and awake rodents [21,22] and was used to study the degree of correlation during oscillatory neuronal activities [23]. However, this approach depends on making the recordings from highly correlated cells, mostly observed in deeply anesthetized animals. Methods for estimation of excitatory and inhibitory inputs of a single cell during single trials were previously developed [24-28]. However, these methods make significant assumptions about the dynamics and statistics of the inputs. Importantly, all these methods rely on the occurrence of membrane potential fluctuations when estimating excitatory and inhibitory conductances. Clearly, changes in conductance sometimes are not accompanied by any change in membrane potential, as expected when a cell receives tonic shunting synaptic input with a reversal potential near the resting potential of the cell.

We describe a new theoretical framework for simultaneously measuring both excitatory and inhibitory conductances under current clamp in a single trial with high temporal resolution, without making statistical assumptions about the inputs. It is based on frequency analysis of the response of neurons when injected with a current composed of two sinusoidal components and allows measuring both the excitatory and inhibitory conductances simultaneously with membrane potential as a function of time. We demonstrate this method in-silico using simulations of a point neuron receiving excitatory and inhibitory synaptic inputs as well as in a realistic pyramidal cell model when synapses are distributed further away from the soma. Finally, we describe the limitations of this approach in whole cell patch clamp recordings obtained using contemporary intracellular amplifiers.

Results

Transformation of membrane potential and total conductance to E and I conductances

We sought to develop a method that provides a way to simultaneously measure the excitatory and inhibitory conductances in a single trial with high temporal resolution during current clamp recording. We begin with the membrane Eq 1 for passive synaptic inputs of a point neuron, which can be rearranged to isolate the excitatory and inhibitory conductances as shown in Eq 2.

Replacing V(t)−V, V(t)−V, V(t)−V with V(t), V(t), V(t) respectively and assuming that the total conductance equals the sum of the inhibitory and excitatory conductance g(t) = g(t)+g(t) we get:

Eq (2) shows that the two inputs can be isolated if the following parameters are known: V(t), membrane voltage; g, leak conductance; g(t), total synaptic conductance; V, V, V, equilibrium potentials of the individual conductances; C, membrane capacitance; I, stimulus current. Fig 1 shows how this equation works in a simulated point neuron where these parameters are indeed known. We demonstrate this transformation by showing depressing excitatory and inhibitory inputs as well as a step change in conductance. However, it works for any type and dynamic of excitatory and inhibitory inputs.

Fig 1

Ge and gi can be obtained from V(t) and total g.

A. Simulation of correlated excitatory (green) and inhibitory (red) synaptic inputs (inhibition delayed by 4 ms after excitation), which depressed according to a mathematical description of short-term synaptic depression (STD, Markram and Tsodyks, 1997). These are the inputs the method aims to reveal. B. Membrane potential simulation of a passive point neuron (R = 300MΩ, C = 0.15nF, Euler method, dt = 0.0005s) receiving the inputs in A, with the total conductance shown below. We assume that these two vectors are measurable. A short test current pulse was injected at the early part of the trace. C The result of transforming V(t), its derivative (not shown) and the total synaptic conductance into ge and gi using Eqs (1 and 2).

How do we find these parameters under experimental conditions? The equilibrium potentials are generally assumed to be known and determined from intracellular and extracellular ion concentrations. The leak conductance and membrane capacitance can be measured when injecting hyperpolarizing current steps. The voltage is also easy to resolve during the current clamp. However, developing a method to record the membrane potential and at the same time also measure the conductance at each time point has been challenging. As we describe below, we can theoretically estimate the total conductance of the cell by measuring the voltage response during injection of a current composed of two high-frequency sinusoidal components. We start with impedance analysis of passive circuits representing a simplified point neuron with a patch clamp pipette and describing the relationships between the impedance and cell conductance.

Impedance-conductance relationship in a passive point neuron

To develop a method that can be practically used for whole cell patch recordings, we included the resistance of the patch pipette in our analysis. As shown below, the resistance of the electrode affects the measurement of the cell’s impedance and thus cannot be ignored. We analyzed in the frequency domain the impedance of a circuit composed of a recording electrode (R) and a simplified point neuron (composed of a conductance, g(t) (equal to g+g(t)+g(t)) and a capacitor, C). The impedance of this circuit is given by Eq 3 (w = 2πf, j is the imaginary unit and f is the frequency in Hertz). The cell conductance (g(t)) and the pipette resistance (R(t) can vary over time, and so consequently also the impedance of the circuit (Z(t)).

Fig 2 illustrates the relationships between the impedance and g for various frequencies (for constant values). It also shows that in the presence of R, impedance-frequency curves intersect each other as frequency increases, resulting in a positive relationship between circuit impedance and g for a large range of g (compare Fig 2A and 2C). The presence of R also keeps the phase almost constant for different frequencies and g values Fig 2D). Thus the electrode resistance has a prominent effect on the total impedance of this circuit and should not be ignored when injecting high frequency sinusoidal current into cells.

Fig 2

Impedance frequency-curves of passive electrical circuits for different conductances.

A. Absolute impedance as a function of frequency for different values of the model conductance. Note that none of the curves intersect. B. Absolute impedance curves as function of conductance together with phase curves between real and imaginary parts of the impedance. Each line represents a different frequency (50Hz to 500Hz, steps of 50Hz, from lowest (pale blue or red) to highest (deep colors) as indicated by the text (Fr = [50:50:500]) above. Also presented are phase curves between voltage and current for the same frequencies. C-D. The same but when the RC circuit is also connected in series to a resistor (Rs). Note in c that curves intersect each other at high frequencies and in d that the phase is almost constant. Fixed circuit parameters: Rs = 30MΩ, C = 0.15nF.

The in-silico experiment

In the next sections, we show the response of a point neuron to an injection of a current (Fig 3D) composed of two sinusoidal components (Eq (4), w = 2πf, w = 2πf): can be used to measure changes in excitatory and inhibitory conductances imposed on the model (Fig 2B) in a single trial. Although the voltage response in our simulation fluctuates across a large range of more than 35mV (Fig 3C), most of the drop of voltage occurs on the electrode resistor, as seen when we set R to zero (Fig 3E). Due to the low-pass filtering of the input by the passive properties of the cell when injecting high frequency sinusoidal current, the fluctuations of the voltage across the membrane itself are extremely attenuated, resulting in less than 6mV peak to peak amplitudes. Such small fluctuations are unlikely to recruit any voltage-gated intrinsic current. Note that the value of the electrode resistance accounts for both the pipette and access resistance. In our our simulation we set the electrode resistance to 30MΩ, which is higher than for the typical access resistance in in-vitro recordings, but well within the range of in-vivo recordings [29]. The current and the voltage are used to calculate all the passive properties of the simulated cell in a single trial (i.e., R(t), g(t) and C). The computations are all analytical and approximation is done only when estimating the cell’s capacitance as shown below. As described above, estimating the cell’s conductance allows us to measure the excitatory and inhibitory conductances.

Fig 3

Measurement of total impedance in a single trial -simulation of a point neuron.

A. Simulated membrane potential of a point neuron when receiving synaptic conductances as shown in B. (excitation–green, inhibition–red). C-D. The voltage response of a simulated neuron I receiving synaptic inputs described in b and injected with a current composed of two sinusoidal components (d, 0.375nA, 210Hz and 315Hz). ‘Recording’ was made via an electrode of 30MΩ and thus most of the voltage drop due to the injected current occurred across the electrode. E. The actual voltage change across the membrane was small (as ‘recorded’ when electrode resistance was set to zero). F-I. voltage and current traces when filtered at the two frequencies used to compose the current. Note for the small fluctuations in voltage. J Impedance curves for each of the two frequencies obtained by dividing the Hilbert transform of the voltage and current shown in F-I and then taking the absolute values. Edge effect of the filtering is observed near zero and end times of the traces.

Measurement of the cell’s total conductance

The first step towards measuring the cell’s excitatory and inhibitory conductances using injection of sinusoidal currents is to measure its total capacitance. The cell’s capacitance is usually estimated from the response to a step current. Other methods for such estimation are also available, such as using a short pulse [30,31] and variance analysis of the response to injection of noise [32]. Here we show that a cell”s capacitance can be well estimated from the response to either one of the two frequencies composing the sinusoidal current (Eq (4)). We rely on the assumption that when the frequency of the current is high (w * C >> g(t)), we can neglect g(t) in the denominators of the second and third terms in Eq 3. Hence, at such frequencies the electrode resistance (R) is relatively larger than the second term, and thus the second term can be neglected. In this case, the total impedance of the circuit is mostly determined by the electrode resistance and the capacitance of the cell, as the latter draws most of the sinusoidal current that is injected into the cell. Here we ignore any stray capacitance in the recording system, such as of the recording pipette, but below we show that this capacitance can be partially compensated offline. The capacitance of the cell can be estimated from the voltage amplitude and phase relationship between the voltage and the current. These relationships can be approximated by Eq (5) (see also the phase curves in Fig 2D) obtained from Eq 3 when w*C>>g.

For such an estimation to be valid (i.e., deriving Eq (5) from (3)), the frequency of each one of the two current components has to be sufficiently high. For example, for a cell with a mean conductance of 1/100MΩ and total capacitance of 0.15nF, recorded with 10MΩ electrode (R), a ratio of ~88 between (w*C) and g will be obtained at 100Hz. Since the impedance of the second term in Eq (3) for this example is ~1MΩ, much smaller than R (10MΩ), we neglect this term. Thus, the capacitance can be obtained from Eq (5), if we can estimate the electrode resistance and the phase relationship between the current and the voltage. We do this in a single trial when sinusoidal current is injected, by first measuring the electrode resistance (R) from the ratio of the absolute values of the fast Fourier Transform (FFT) of the voltage and the current at the frequency of the injected current, after both traces were bandpass filtered at one of the two frequencies (F1 or F2, using ‘bandpass’ Matlab function, implementing finite impulse response (FIR) filter). Importantly, this calculation is performed for a time window within which no stimulation is delivered (e.g., 1 second before stimulation). The two vectors (FV, FI bandpass filtered voltage and current) are then used to estimate R. For the measurement of the capacitance we provide a rough estimation of R, denoted with an asterisk. A more precise estimation of R is provided later.

The phase between FV and FI is calculated from the Hilbert transform of FV (H operator, either for the F1 or F2) using the ‘hilbert’ Matlab function and averaging over time:

Averaging is performed for the same time window as above, within which no stimulation is delivered (e.g., 1 second before). The trigonometric relationships between the real and imaginary parts in Eq (5) are described in Eq (8), allowing to estimate the cell’s total capacitance given that R and θ are measured as described in Eqs (6) and (7):

In the example shown in Figs 2 and 3, the real capacitance was set to 0.15nF and was estimated as 0.149nF. Note, that estimation of C can also be obtained when setting R to zero at a similar accuracy.

We then use the estimated capacitance of the cell to measure the cell’s conductance and to obtain a more accurate measurement of the electrode resistance, both over time in a single trial. In this computation these values will be measured based on the analytical solution of Eq 3, this time without making any approximations. Here we use the fact that the current contains two sinusoidal components having two different frequencies (F1 and F2, e.g., 210Hz and 315Hz as used in the example). Since Z(f) decreases with increasing frequency (Fig 2), increasing the frequencies, although it allows higher temporal resolution, will reduce the signal to noise ratio in the presence of noise. The voltage and the current are then bandpass filtered at the two frequencies (Fig 3F–3I, due to screen resolution are displayed as patches of colors). Note the small modulations in the bandpass filtered voltage signals, which are in the order of about 1mV. These modulations result from changes in the cell’s conductance during the simulation of the synaptic inputs following the relationships between them as shown in Fig 2. For each bandpass filtered voltage and current trace: FV1(t), FV2(t), FI1(t), FI2(t) we computed thehilbert transforms (HFV1(t), HFV2(t), HFI1(t), HFI2(t), using the ‘hilbert’ Matlab function). These complex vectors are then used to calculate the impedance of the cell at the two frequencies over time:

The absolute values of these complex vectors, shown in Fig 3J, demonstrate curves with a shape that is similar to that of the total conductance of the cell (leak plus synaptic conductances). Note that when the conductance of the cell is increased during activation of these inputs, the impedance is also elevated. This only happens in the presence of R, as shown in Fig 2.

These two impedance vectors are then used together to solve Eq 3 and obtaining a solution for R(t) and g(t) (when z1≠z2, C is the estimated capacitance). To this end we used Mathematica (Wolfram) to solve the two equations for absolute values of z and z (“Solve[Abs (r + 1/(g + I*w1*c)) = = Abs (z1) && Abs (r + 1/(g + I*w2*c)) = = Abs (z2), {r, g}]”, I = imaginary unit in Mathematica (Wolfram)) which gives the following solutions for R and g (here Z and Z are complex time dependent vectors, j is the imaginary unit, and C is the estimated capacitance):

In Eqs (11) and (12) z1, z2 as well as R(t) are time dependent variables. Identical estimation will be obtained in Eq 12 after replacing w1 and z1 with w2 and z2. In Fig 4A, we again plotted the two impedance curves and also included the electrode resistance (R(t)), which is only slightly larger than its real value used in the simulation. The estimated total conductance is plotted in Fig 4C. Note that the estimated total conductance is almost identical in shape and magnitude to the sum of the leak, excitatory and inhibitory conductances used to simulate the membrane potential in this example.

Fig 4

Measurement of ge and gi in a single trial from impedance measurements–- simulation of a point neuron.

A. Absolute impedance curves for each frequency (as shown in Fig 2). B. The membrane potential after filtering out the components at the injected frequencies. C. Measured conductance and electrode resistance (shown in A) were obtained from the absolute impedance curves (Eqs 11, 12) and the estimated capacitance. D. Excitatory and inhibitory conductances were estimated from the cleaned voltage and measured conductance (Eqs 1, 2). Insets show the extended data for the first synaptic responses, superimposed with the real conductances (thin lines) (scale bars are 100ms and 10nS. E. Measured excitatory and inhibitory values plotted against real values that were used for the simulation.

Estimation of the excitatory and inhibitory conductances from cell’s conductance and membrane potential

After estimating the total conductance, Eqs (1) and (2) are used to compute the excitatory and inhibitory conductances as discussed above. Since sinusoidal current is injected into the cell (with two frequency components) we bandstop filter around each frequency (+- 5Hz) to obtain a clean version of the membrane potential. Before we use Eqs (1) and (2), we need to calculate the resting membrane potential and its corresponding leak conductance. We do this by finding the mean voltage in the cleaned membrane potential for the lower 5th percentile of the total conductance vector, which we assume reflects the resting state at which no synaptic inputs are evoked (i.e, g). The corresponding membrane potential values for this 5th percentile conductance were used to calculate the mean resting potential (V). The synaptic conductance is simply given by: g(t) = g(t)−g (the difference between total conductance and leak conductance). In the transformation presented in Eqs (1) and (2), we assume that the reversal potentials of excitation and inhibition are available to us (i.e., 0mV and -70mV). The capacitance and total conductance are obtained as described above. The results of these computations are shown in Fig 4D. Our calculations revealed that the estimated conductances are almost identical to the real inputs of the simulated cell (compare Figs 3B to 4D). We note that our method allows estimating the conductances even when tonic input exits, as demonstrated in the step change in excitation and inhibition (shown between 3 to 4 seconds). In fact, the Pearson correlation between the real inputs and the estimated inputs for this simulated example were extremely high: 0.999 for excitation and 0.996 for inhibition (Fig 3E).

Computing the excitatory and inhibitory conductances of a cell embedded in a balanced network

We asked if our approach can be used to reveal the underlying excitatory and inhibitory conductances of a model cortical neuron embedded in an active network where it receives excitatory and inhibitory inputs. Therefore, we used a simulation of a cortical network at a balanced asynchronous state [33] to obtain the excitatory and inhibitory synaptic inputs of a single cell (kindly provided by Dr. Michael Okun, University of Leicester). We used these conductances in a simulation of a single cell, in which we injected a current with two sinusoidal components (210 Hz and 315Hz) via a 50MΩ electrode and measure the response of the cell, before (Fig 5) and after filtering out the two sinusoidal components from the membrane potential (Fig 5B, black trace, which is superimposed almost perfectly with the one obtained without current injection, blue trace).

Fig 5

Estimation of the correlation between excitatory and inhibitory conductances of a neuron embedded in balanced/unbalanced Networks.

A. Excitatory and inhibitory conductances taken from network simulations (see text) were used to simulate the response of a point neuron recorded with a 50MΩ pipette while injected with 210Hz & 315Hz 0.5 nA sinusoidal current. B. The voltage in A was ‘cleaned’ from the sinusoidal component (black trace) and it is displayed with the voltage response of the cell when no current was injected superimposed with the ‘cleaned’ voltage. C-D. Estimated excitatory and inhibitory conductances superimposed with the imposed conductances of the simulated cell. E, G. Color lines describe the cross-correlations between the imposed and estimated excitatory and inhibitory conductances for the balanced cortical Dynamics. F. The correlations between measured g and g (blue line) and between the imposed g and g (black line). H-J. The same analysis as for e-g but after shifting the inhibitory input by 10 seconds to mimic uncorrelated excitatory and inhibitory inputs.

We then used our computations to estimate the excitatory and inhibitory conductances (Fig 5C and 5D). Note, however, that for both inputs the estimated conductances are more negative than expected. This is simply because the leak conductance was estimated from the 5th percentile of the total conductance of the cell, but since synaptic activity persisted throughout the trace, the leak conductance reflects a mixture of the true leak conductance and some baseline synaptic activity. Nevertheless, the estimated excitatory and inhibitory synaptic conductances were very similar to those used as inputs (Fig 5E and 5G), and similarly to the real inputs, estimated E and I conductances were highly correlated (Fig 5F). Our approach was also successful in measuring E and I inputs when they are not correlated (Fig 5H and 5J, by shifting the inhibitory input by 10 seconds relative to excitation). Indeed, as expected for this case, no correlation was measured between the measured inputs (Fig 5I). In summary, our approach allows accurate estimation of excitatory and inhibitory inputs in various conditions without any need to take into account the dynamic and statistical properties of the excitatory and inhibitory inputs.

Measurement of E and I inputs during large variations in access resistance

Changes in access resistance due to incompletely ruptured membrane or other due to movement of the recorded cell and preparation, pressing the pipette onto the membrane, are well-known limitations of whole cell patch recordings. However, one of the advantages of the approach is in its ability to track changes in the electrode and access resistance and taking them into account when calculating the total conductance of the cell with a high temporal resolution. We demonstrate it by simulating rapid changes in the electrode resistance during the in-silico recordings (Fig 6, identical synaptic inputs to those used in Fig 3). These variations led to a noisy impedance measurement (Fig 6A). However, since we can measure the access resistance over time (R(t)) and total g(t) at the same time (Eqs (11) and (12)), followed by measurement of excitation and inhibition as described in Eq (2), the changes in the electrode resistance had no apparent effect on the ability to accurately estimate the inhibitory and excitatory conductances (Fig 6D and 6E).

Fig 6

Measurements of ge and gi are accurate even when electrode resistance is not stable.

A. Absolute impedance curves for each frequency depicted together with measured electrode resistance. To modulate the electrode resistance, we smoothed a lowpass Normally distributed noise signal and added it to a fixed resistance. All other parameters of the inputs are identical to those shown in Figs 3 and 4. B-E. Analysis of the voltage response of the point neuron as processed by the same way as shown in Figs 2 and 3. Note for the accurate estimates of ge and gi (compare to Fig 3B).

Measurement of E and I inputs in the presence of realistic noise

Next we asked how sensitive our measurements are in the presence of realistic noise. Therefore, we used a typical patch electrode to record a voltage trace in a slice setup when positioning the electrode outside a neuron (kindly provided by Dr. Alexander Binshtok, Hebrew University). We then added this noise to our simulated voltage prior to the measurement of excitation and inhibition (Fig 7). A sample of the voltage in the absence of sinusoidal current injection is shown in the inset of Fig 7A (voltage scale bar is 0.5 mV). Despite the presence of such noise (standard deviation of 0.04mV), and a concomitantly noisier measurement (Fig 7D) of excitation and inhibition, their values closely matched those we imposed as inputs in the simulation (Fig 7E).

Fig 7

Measurements of ge and gi are accurate in the presence of realistic recorded noise.

A. Absolute impedance curves for each frequency depicted together with the measured electrode resistance when real noise recorded from an in-vitro step was added to the simulated voltage before the measurement. The inset shows a voltage trace in absence of sinusoidal injection (scale bars: 0.5s, 0.5mV). All other parameters of the inputs are identical to those shown in Figs 3 and 4. B-E. Analysis of the voltage response of the point neuron as processed in the same way as shown in Figs 2, 3 and 6. Refer to Fig 3B for ground-truth ge and gi.

Compensation for electrode capacitance

In the above computations we assumed that the recordings are made with a pipette of zero capacitance. However, electrode capacitance can greatly affect the measurement using our novel algorithm. Most of the stray capacitance of recording pipettes is formed by the separation of the solution inside vs. outside the glass pipettes. Experimentally, it can be reduced but not eliminated by coating the pipette with hydrophobic material [34]. Pipette capacitance (C, illustrated in Fig 8) can also be neutralized by the electronic circuit of the intracellular amplifier, using a positive feedback circuit. In our in-silico experiment, we show that C can greatly affect the measurement, as pipette capacitance draws some of the injected sinusoidal current. As a result, the impedance measurements for the two frequencies (z1 and z2) are smaller than expected from the cell and R alone (Fig 8B, Rs is 20MΩ and the curves are well below this value). This, in turn, results in a much higher leak conductance and a completely wrong estimation in the synaptic conductances based on Eqs (11) and (12). Altogether, our estimations can be flawed, leading to negative evoked inhibitory conductance (Fig 8D).

Fig 8

Computational approach for compensation of parasitic capacitance for measurement of excitation and inhibition in an in-silico model.

A. The ‘real’ inputs in the in-silico experiment and the circuit representing the recording configuration. Note for Cp, the parasitic capacitance of the electrode (Rs). B-D. Measurement of the impedance, followed by calculation of the conductance and E and I inputs using the approach described in Figs 3 and 4. E-G. Similar measurements when recalculating the impedance at each frequency using Eqs (13) to (18) to subtract the effect of pipette capacitance.

To compensate for the impedance reduction due to the pipette capacitance we estimated Cp and then used this value to correct the measured impedances. Here we show the theoretical admittance (Y, Y = 1/Z) at each of the two frequencies for the equivalent circuit of a cell recorded with a pipette that has stray capacitance, as shown in Fig 8. The second terms in the following Equations depict the admittance of the stray capacitance (Eqs (13)–(15) were derived from the circuit that is presented in Fig 8, G is the cell’s total conductance).

From these two equations and replacing with Y (and Y), C is given by:

However, the value of Y and Y are unknown and are those we seek. We found, however, that the second term (Y-Y) can be neglected as it is much smaller when compared to the value of 1/Z1−1/Z2. For example, for the parameters used in this simulation, the ratio between the latter and first terms is ~200, clearly justifying our next approximation in which we use in the measured impedance curves, as made using Eqs (9) and (10) (shown as measured Z1 and Z2 below, both are time dependent).

We then use this estimated value of C (averaged for a selected time window (e.g., 1S) before the stimulation under the assumption that synaptic inputs are silent during this time) to calculate the estimated impedance of the cell and the electrode alone, as theoretically expected () which is done by subtracting from the two measured Z curves the C component following rearranging Eqs 13 and 14:

The new Z′ vectors are then used as the inputs as described above in Eqs (11) and (12) and the subsequent process as described above. This approach greatly improved the measurement of excitation and inhibition (Fig 8E–8G). Hence, this component in the analysis, which can be switched on and off, can help resolve the analysis of real recordings, where stray capacitance always exists.

Measuring synaptic conductances in morphologically realistic neurons

To assess how our method resolves dendritic conductances, we simulated a morphologically realistic CA1 pyramidal cell [35]. We uniformly distribute 50 inhibitory and 50 excitatory synapses proximal to the soma. We realized that due to current escape of the injected sinusoidal current to the dendrites, the estimated leak conductance is much larger than its actual value. In the case of proximal synaptic inputs, less current is escaping towards the dendrites during activation of these inputs when compared to pre-stimulation conditions. We compensated for this change by dynamically altering the strength of the leak conductance at each time point based on the estimated total synaptic conductance before calculating the excitatory and inhibitory conductances (Eqs (1) and (2)) by using this empirical equation:

Such change is equivalent to a dynamic change in the electrotonic length of cells, known to cause space clamp errors [36-38]. It shows that for weak proximal synaptic input this function strongly reduces the newly calculated leak conductance (g′(t)) as expected, and that this allows to compensate for the current escape. However, when the synaptic inputs get stronger the function increases the leak, as less current is expected to escape to the dendrites due to the shunting effect of the input.

Although those synapses are on average 129.92μm (±47.83μm SD) away from the soma, our method resolves the excitatory and inhibitory conductances in a single trial at least as well as the voltage clamp measurements do during two separate trials. When the synapses are moved further away, to an intermediate distance of 238.69μm (±39.71μm SD), our method underestimates the conductance to the same extent as voltage clamp (Fig 9B). Under most biological conditions synapses are not constrained to a narrow part of the dendrite. Therefore, we uniformly distributed synapses anywhere on the apical dendritic tree (Fig 9C). This resulted in synapses with an average distance to the soma of 309.92μm (±164.46μm SD). In this case, our method still follows the conductances but underperforms compared to voltage clamp. Because the measurement quality seemed to decrease with distance, we did more simulations to quantify the relationship between somatic distance and recording quality.

Fig 9

Measuring conductance changes of dendritic synapses.

A. Inhibitory and excitatory synapses were placed proximal (129.92μm ±47.83μm SD) to the soma and all measurements were performed at the soma. Simultaneous conductance measurements are at least as accurate as separate voltage clamp recordings for these proximal synapses. B. Inhibitory and excitatory synapses were placed at intermediate distance (238.69μm ±39.71μm SD). Simultaneous conductance measurements and voltage clamp are both well correlated with the temporal dynamics but underestimate the amplitude. C For distributed synapses (309.92μm ±164.46μm SD) simultaneous conductance measurements underestimate the magnitude of the true conductance but still follow the time course.

Conductance measurements of proximal inputs are stable and reliable

To investigate the relationship between measurement quality and synaptic distance to soma, we simulated a single excitatory and a single inhibitory synapse at the same dendritic segment. As above, we found that we can reliably isolate the conductances when the synapse pair is close to the soma (Fig 10A). At an extremely distal synapse localization, the measurement becomes unreliable. Even the voltage clamp ceases to follow the temporal dynamics. To quantify the extent to which our measurement follows the temporal dynamics of the current we calculated the correlation coefficient between measurement and true conductance. We found that the measurements are very reliable for synapses below 400μm somatic distance (Fig 10C). Above that distance, the measurement quality breaks down abruptly for the excitatory conductance (Fig 10B and 10C).

Fig 10

Simultaneous conductance measurements are accurate for synapses up to 400μm but break down above.

A. Simultaneous conductance measurements are highly accurate for the excitatory and inhibitory proximal synapses. B. For extremely distal synapses, our simultaneous conductance measurements become inaccurate. Color legend as above. C. The correlation coefficient for many synapses confirms that simultaneous conductance measurements are at least as accurate as voltage clamp below 400μm somatic distance. For further away synapses, the conductance measurement technique breaks down abruptly for the excitatory conductance.

Discussion

We describe a novel framework to estimate the excitatory and inhibitory synaptic conductances of a neuron under current clamp in a single trial with high temporal resolution while tracking the trajectory of the membrane potential. We show that the method allows estimating these inputs also in a morphologically realistic model of a neuron. The work described above here is theoretical and lays the foundations for future experimental work.

The method is based on the theory of electrical circuit analysis over time when a cell is injected with the sum of two sinusoidal currents. This allows us to measure excitatory and inhibitory conductances and at the same time track the membrane potential.

We demonstrated the method in simulations of a point neuron and in realistic simulations of a pyramidal cell, receiving proximal and uniformly distributed synaptic inputs. For the point neuron, we showed that we could reveal the timing and magnitude of depressing excitatory and inhibitory synaptic inputs with high temporal resolution and accuracy of above 99% (Figs 3 and 4). In another example, we used our method to reveal these inputs during an asynchronous balanced cortical state and showed that excitation and inhibition dynamics can be measured with high accuracy. Importantly, these estimations were obtained from single trials and allowed obtaining the natural dynamics of the membrane potential by filtering out the sinusoidal components of the response to the injected current. Therefore, our method is especially suitable for estimation of excitation and inhibition when these inputs are not locked to stereotypical external or internal events, such as during ongoing activity. We note that when injecting high frequency current (of a couple of hundred Hertz and above), the voltage drops mostly across the recording electrode. Here we tuned the current amplitude to produce a few millivolts sinusoidal fluctuation across the cell membrane, which should have minimal effect on voltage-dependent intrinsic and synaptic conductances when performing recordings in real neurons.

Comparisons with other methods

Measurement of average excitatory and inhibitory conductances of single cells: Excitatory and inhibitory synaptic conductances of a single cell were measured both under voltage clamp or current clamp recordings, focusing in-vivo on the underlying mechanisms of feature selectivity in sensory response of cortical cells and on the role of inhibition in shaping the tuned sensory response of mammalian cortical neurons [6,8,39]. Conductance measurement methods were also used to reveal the underlying excitatory and inhibitory conductances during ongoing Up and Down membrane potential fluctuations, which characterize slow-wave sleep activity [40,41]. The advantages and caveats of these methods were reviewed in [29]. Common to these conductance measurement methods is the requirement to average the data over multiple repeats, triggered on a stereotypical event (such as the time of sensory stimulation or the rising phase of an Up state) and then average trials at different holding potentials. The averaged data is then fitted with the membrane potential equation (assuming that the reversal potentials are known) to reveal the conductance of excitation and inhibition at each time point. However, these methods cannot reveal inhibition and excitation simultaneously in a single trial, and only estimate averaged relationships. Our proposed method, on the other hand, allows for simultaneous measurements during a single trial. Importantly, since there is no need to depolarize or hyperpolarize the cell, our method allows measurement of synaptic conductances at the resting potential of the cell, potentially obtaining measurements of voltage dependent conductances as they progress during the voltage response to the synaptic inputs. We note that our method shares the basic approach for the analysis of point-neurons using the theory of frequency analysis of electrical circuits with capacitance measurements methods [42,43].

An alternative approach for estimating the excitatory and inhibitory conductances of a single cell was demonstrated for retinal ganglion cells [44]. In this study the clamped voltage was alternated between the reversal potential of excitation and inhibition at a rate of 50 Hz and the current was measured at the end of each step. This study revealed strong correlated noise in the strength of both types of synaptic inputs. However, unlike the method proposed here, the underlying conductances are not revealed simultaneously and–due to the clamping–the natural dynamics of the membrane potential is entirely unavailable, preventing examining the role of intrinsic voltage dependent dynamics in the generation of neuronal subthreshold activity.

Single trial measurements of ge(t) and gi(t) under various assumptions on synaptic dynamics

Theoretical and experimental approaches based on the dynamics of excitatory and inhibitory conductances in a single trial were previously proposed. Accordingly, excitation and inhibition are revealed from current clamp recordings in which no current is injected. Approaches based on Bayesian methods which exploit multiple recorded trials were proposed [25] and estimation of these inputs in a single trial were also proposed but lack the ability to track fast changes in these conductances [24]. A group of other computational methods [26-28] showed that excitatory and inhibitory conductances could be revealed in a single trial when analysing the membrane potential and its distribution. Common to all these methods is the requirement to observe clear fluctuations in the membrane potential. Our method, however, allows revealing these inputs even if no change in membrane potential due to synaptic input is observed (except for the response to the injected sinusoidal current). Changes in conductance are often expected even when the membrane potential is stable, for example when a cell is receiving tonic input (see the step change in excitation and inhibition in Figs 3 and 4, between 3 to 4 seconds, resulting in a constant membrane potential value) and when a constant balance in excitatory and inhibitory currents exists.

Paired intracellular recordings

The substantial synchrony of the synaptic inputs among nearby cortical cells [19,21,45,46] allows continuous monitoring of both the excitatory and inhibitory activities in the local network during ongoing and evoked activities. A similar approach was also used to study the relationships between these inputs in the visual cortex of awake mice [22] as well as gamma activity in slices [4]. While paired recordings are powerful when examining the relationships between these inputs in the local network, such recordings do not provide definitive information about the inputs of a single cell. Moreover, although the instantaneous relationship between excitatory and inhibitory inputs can be revealed by this paired recording approach, the maximum inferred degree of estimated correlation between excitation and inhibition is bounded by the amount of correlation between the cells for each input, which may change across stimulation conditions or brain-state [47-49]. For example, a reduction in the correlation between excitation, as measured in one cell, and inhibition measured in the other cell can truly suggest smaller correlation between these inputs for each cell, but it can also result from a reduction of synchrony between cells, without any change in the degree of correlation between excitation and inhibition of each cell. This caveat of paired recordings prevents us from finding, for example, if cortical activity shifts between balanced and unbalanced states [50,51]. Simultaneous measurement of excitatory and inhibitory conductances of a single cell across states will allow these and other questions to be addressed.

Limitations

Theoretically, increasing the frequency of the sinusoidal waveforms of the injected current in our method improves the temporal precision when measuring synaptic conductances. However, this comes at the expense of sensitivity, which reduces as frequency increases (Eq 3 and Fig 2). In our simulations we limited the frequency of the injected current up to about 350Hz. At this range, our simulations, depicting realistic passive cellular properties and typical sensory evoked conductance will result in a clear modulation in voltage when injecting ~1nA sinusoidal current. When bandpass filtering the voltage, the modulation is in the order of only a mV, but is still above the equipment noise.

We show that changes in access resistance due to incompletely ruptured membrane or other factors, such as mechanical vibration causing the membrane to move with respect to the pipette, can be well measured and compensated (Fig 6). Hence our approach can be implemented to estimate excitatory and inhibitory inputs of a cell in these realistic conditions.

Another aspect that might reduce the sensitivity of our method is the presence of pipette stray capacitance. We developed a modular component in the analysis that can be used to correct some of this stray capacitance (Fig 8). Importantly, no additional measurement is needed beyond the injected sine waves, done in a single trial, to measure this stray capacitance and compensate for its effect. Yet, when stray capacitance is much higher than was demonstrated here, this approach fails to provide a good estimation of the synaptic conductance. Hence, special care will still be needed to minimize any stray capacitance as much as possible.

We demonstrate in simulations of morphologically realistic neurons that we can estimate proximal synaptic inputs in a single trial using our approach. Although we underestimated these inputs when compared to simulated voltage-clamp experiments, their shape and relationships were preserved in our measurements if the inputs impinged on dendrites not more distant than 400 μm from the soma of our implementation of a pyramidal cell. Even though this limitation should be considered in real recordings, these data also suggest that the method will provide an adequate assessment of proximal inputs.

Possible application of the method for measurement of non-synaptic intrinsic conductances

Our method can also be used when voltage-dependent conductances evolve naturally, as we can measure these inputs at the resting potential of the cell, as long as the sinusoidal fluctuations across the membrane due to the injected current are small. Such an approach therefore can be used when performing pharmacological tests, such as testing effects of modulators, agonists and antagonists of various ion channels. Due to the ability to measure these inputs in a single trial, the time course of the effects can be studied in rapid time scales while examining the effects of such drugs on both inputs at the same time.

In summary, our theoretical study shows that synaptic and other conductances can be measured at high temporal resolution in a single trial when cells are recorded at their resting potential. More research is needed to find if this approach can be used successfully during physiological recordings from real neurons.

Feasibility of the technique in real recordings

The expected signal to noise ratio, based on the addition of realistic noise (Fig 7) is sufficiently high to measure the excitatory and inhibitory input during in-vitro recordings. However, it is clear that this framework has to be tested in real recordings of neurons. We fully disclose that we made attempts to test the method in real recordings and discovered that in most of our recordings, none shown here, measurements were unsuccessful. Following tests for impulse response of the amplifier, we found that this results from an active feedback circuit in our intracellular amplifiers. We are currently improving the amplifier circuitry and in parallel developing algorithms that will incorporate the frequency response characteristics of these amplifiers.

Methods

Simulations

To develop the method we constructed a simple simulation of a single compartment neuron attached to a resistor, simulating the resistance of the recording pipette (R is the electrode resistance). I is the injected current and the other variables as shown in Eqs (1) and (2). Also note that the capacitive current is given by: I = I−k·(V−V)/R, where V is the recorded voltage (across the recording the pipette), V is the voltage across the membrane only I is stray current. For k = 0 we assume no stray capacitance and for k = 1, capacitance was included. Hence at each time point we calculated (dt is the time step of the simulation):

To test the performance of our method in extraction of excitatory and inhibitory conductances, we simulated the response of a cell to a train of synaptic inputs which depress according to the mathematical description of short term synaptic depression (STD, [52]) with τ = 0.003S (inactivation time constant) for excitation and τ = 0.5S (recovery time constant) for excitation and the same inactivation time constant for inhibition (0.003S) but a longer recovery time constant (τ = 1.3S) but exhibiting the same utilization (0.7). The values of the passive properties of the cell and the strengths of synaptic conductances in the simulation were chosen to be at a similar range of experimental data [8,14,15]. Namely, resting input resistance of 150MΩ, total capacitance of 0.15nF and pipette resistance of 30MΩ. Simulations were run using a simple Euler method with a time step of 0.1msfor all point neuron simulations except for Fig 7 (0.025ms).

Morphologically realistic simulations

We used NEURON 7.6.7 [53] in Python 3.7.6 to simulate a CA1 pyramidal cell [35]. We loaded this cell directly into NEURON without changes to the neuron model. 50 inhibitory and 50 excitatory were distributed on parts of the apical tree. The synaptic mechanism was a modified version of the Tsodyks-Markram synapse [52] where we added a synaptic rise time (NEURON mechanism available at https://github.com/danielmk/ENCoI/tree/main/Python/mechs/tmgexp2syn.mod). The synaptic parameters are detailed in Table 1. Event frequency of both synapses was 10Hz and events were jitter with a Gaussian distribution of 10ms SD.

Table 1

Synaptic parameters of morphologically realistic simulations in Figs 9 and 10.

Parameters for Fig 9
Parameter	Value Excitatory	Value Inhibitory	Description
n_syn	50	50	Number of excitatory synapses
gsyn	3e-4μS	4.5e-4μS	Synaptic weight
tau_rise	1ms	1.2ms	Rise time constant of excitatory conductance
tau_decay	20ms	100ms	Decay time constant of excitatory conductance
tau_facil	0ms	0ms	Facilitation time constant
tau_rec	200ms	600	Recovery time constant
U	0.2	0.4	Utilization constant synaptic efficacy
Ev	0mV	-75mV	Reversal potential
Parameters for Fig 10
Parameter	Value Excitatory	Value Inhibitory	Description
n_syn	1	1	Number of excitatory synapses
gsyn	3e-2μS	4.5e-2μS	Synaptic weight
tau_rise	20ms	30ms	Rise time constant of excitatory conductance
tau_decay	50ms	100ms	Decay time constant of excitatory conductance
tau_facil	0ms	0ms	Facilitation time constant
tau_rec	200ms	600ms	Recovery time constant
U	0.2	0.4	Utilization constant synaptic efficacy
Ev	0mV	-75mV	Reversal potential

All measurements were performed at the soma. To simulate an access resistor in current clamp we added a section with a specified resistance between the current clamp point process and the soma. The access resistance was 10MOhm. For the stimulation current we summed two sine waves of 210Hz and 315Hz. The combined sine waves had a peak-to-peak amplitude of 1nA. Voltage clamp was performed in separate simulations with 10MOhm access resistance as during current clamp. While isolating the excitatory current, we clamped at the reversal potential of inhibitory synapses (-75mV). While isolating the inhibitory current, we clamped at the reversal potential of excitatory synapses (0mV). To convert current to conductance, we divided the current by the clamped voltage minus the synaptic reversal potential.

To investigate the relationship between measurement quality and dendritic path distance to soma, we moved a single excitatory and a single inhibitory synapse to the same dendritic section. Sections were chosen by iterating through the list of apical dendrites in steps of 5. The synaptic parameters are detailed in Table 1.

Python simulation results were saved as.m files using SciPy [54]. Simultaneous conductance analysis and plotting were performed in MATLAB.

4 Jul 2021

Dear Ilan Lampl

Thank you very much for submitting your manuscript "A novel theoretical framework for simultaneous measurement of excitatory and inhibitory conductances" for consideration at PLOS Computational Biology.

As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. In light of the reviews (below this email), we would like to invite the resubmission of a significantly-revised version that takes into account the reviewers' comments.

Although all the reviewers were generally excited about the novel method for extraction of conductances of excitation and inhibition presented in the manuscript, there were a number of major issues. Both reviewer 2 and 3 suggested that the manuscript, being a methods paper, needs testing with experimental data, and perhaps be more suited for an experimental journal rather than the computational scope of the PLOS comp biology. Based on these issue we unfortunately cannot accept your manuscript for publication in PLOS comp biology at this point. I hope you will find the comments useful.

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Reviewer's Responses to Questions

Comments to the Authors:

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Reviewer #1: Comments to the Authors

In this paper, authors provided a novel method to simultaneously extract excitatory and inhibitory synaptic conductances of a single neuron from the recorded membrane potential at a single trial. The method is analytical and looks very interesting. It uses two high frequency sinusoidal

Components (as injected current) and solves the general equation of subthreshold membrane potential of a single neuron for the total conductance (Gs) and the electrode resistor (Rs). From these items, and also estimating the cell’s capacitor, the proposed method infers Ge and Gi as excitatory and inhibitory synaptic conductances, respectively. To me this method is very novel. However, I vote to reject just because this method has the potential to be improved and validated with experimental data. Having said that, I should highlight that the authors provided very nice simulations to demonstrate the accuracy of the method. As well, all the points in the Discussion are appreciated. I have some major points as follow.

This method should be tested with experimental data

What is the impact of observation noise on the accuracy of estimated Ge and Gi?

How the clean voltage is calculated? Is it filtered again?

If I understood correctly, the idea of injecting two sufficiently high frequency injected currents was for estimating Gs and Rs (as well as C). Although I really liked this idea, I do not understand why authors do not inject more than two currents, for example 5 high frequency currents, and solve those conductances using a simple optimization technique (without any assumption or simplification).

Reviewer #2: # Comments to the Authors

Authors propose a frequency-domain method to infer simultaneously the excitatory and inhibitory (summated) time-varying conductances, from a single intracellular neuronal recording. As opposed to existing methods, Authors first simulate a single-trial current-clamp stimulation in a a passive RC circuit, while recording its electrical potential. They later explore the performance of the method in another numerical simulations, employing a multicompartment biophysically realistic CA1 model cell.

I disclose that I must have seen an earlier and more primitive exposition of this idea as a 2019 bioarXiv entry. It greatly captured my interest as an experimentalist. I think that the concepts behind the method are definitely of immediate interest for the (more) specialised audience of a methods journal (e.g. Neuron, J. Neurosci. Method) - even though the method has not (yet) been demonstrated by the Authors in an experiment - and less for the audience of a Comp Neurosci journal.

I recommend major revisions.

## Major points

1)

The presentation of the manuscript could definitely be improved. Math notation and precision in using throughout the text is not "stable" and should be revised carefully.

Especially the description of the math could be made a bit clearer, taking a bit more of presentation, discussion and analysis. Some of the steps skipped (definitely accessible for a reader with a math or engineering background) might make a naive reader lost.

2)

I recommend an improved exposition of the hypotheses behind the use of frequency-domain method, with quasi-stationary parameters. Strictly speaking, eq. 3 is obtained from eq. 1, upon Fourier transformation, but a sort of "separation of time-scales" has been invoked, adopted, but not illustrated. While Fig. 2 is obtained keeping g_e and g_i fixed in time, eq. 3 assumes they can vary. If they do vary with time, then Fourier-transforming eq. 1 is no longer straightforward in the frequency domain.

I see of course that this does work in simulations, as a reverse method, but it still lacks of a justification and a more extensive discussion.

3)

The choice of the sample values of the electrode resistance (30 and 50 MOhm) seems unusual at a first sight. Could it be that it includes the access resistance? This is mentioned nowhere and it seems anyway not typical in "good" (in vitro) recordings. Perhaps do Authors refer to access resistance in vivo? Please explain and give some references to the literature.

Does your method for estimating the capacitance have similarities with this classic 1988 paper by Neher (10.1007/BF00582306) and this 1997 extension (10.1016/S0006-3495(97)78810-6), using two sine waves ? A discussion of similarities and differences with past approaches might be useful for the curios reader.

Could the approach of Badel et al. (10.1007/s00422-008-0259-4) be of any relevance for the capacitance estimation (see their Fig. 1c)?

4)

Can you comment on the opportunities/obstables represented by online parametric (concentional bridge balance and capacitance neutralisation circuitry) and (offline) non-parametric (as in the Active Electrode Compensation of Brette and colleagues - see 10.1016/j.neuron.2008.06.021)?

Particularly the latter method should (theoretically) capture the impulse response of pipette and amplifier. That might ameliorate your implementation problems.

5)

I would make it more explcit that you are no longer considering any off-sets in the membrane potential and that all the quantities are referred to the "effective" resting membrane potential.

However this "effective" resting potential is, strictly speaking, not constant as it is equal to:

(gl Vl + gE(t) Ve + gI(t) Vi) / (gl + gE(t) + gI(t))

Can you clarify if and why these effects are negligible in your hands?

6)

An expansion of the discussion on "Feasibility of the technique in real recordings" is strongly suggested.

As the Authors are aware, this is the strongest selling point of the method.

Some aspects related to (old) patch-clamp amplifier designs criticism (see 10.1016/S0006-3495(98)74007-X and 10.1016/s0166-2236(96)40004-2) might be perhaps of use.

## Minor points:

1)

I suggest using (in general) "j" for the imaginary unit, to distinguish it from the current.

I also suggest expressing eq. 3 as

Z(w) = R_s + 1/(G_s + j w C)

rather than rearranging terms to get real_part + j * imaginary_part. It will resonate immediately with people familiar with RC filters. Calculation of magnitude and phase can be anyway done quickly by the properties of the "ratios and products of complex numbers".

2)

Throughout your manuscript, you considered a rather strong current injection (~1-2 nA large). I would make it clear that these are not DC stimuli, leading certainly to spiking activity. They are AC stimuli, like those employed for the analysis of subthreshold membrane resonance (see the classic work by Nelken and colleagues).

It is important to stress that the frequency is so high (compared to the membrane subthreshold cut-off frequency) that the membrane potential will be extremely attenuated and thus (almost surely) not recruiting any voltage-gated intrinsic current.

3)

The math notation is (very) often non-uniform throughout the text, or a bit imprecise.

- page 4, eq. 1 and line 8: please use (V_l, V_e, V_i) or (V^l, V^e, V^i), not both.

- page 4, eq. 1, please make it clear that g_e and g_i (and g_s) are, together with V, all functions of time: make the "(t)" dependency explicit.

- page 4, eq. 1: although conventional, the current "I" should have a positive sign on the right hand side of the equation, assuming as positive the current "entering" the cell;

- use, throughout the manuscript and figures, "f_1" and "f_2" (subscripted indexes) for the frequencies instead of "f1 and f2"sometimes and "F1 and F2" some other times.

- use, throughout the manuscript and figures, "g_e" and "g_i" (subscript), instead of sometimes using "G_e" and "G_i".

- page 7 (text and eq. 4): please use subscript for pulsations "w1", "w2", and frequencies "f1","f2", as you do it for the two components amplitude (i.e. I_1 and I_2).

4)

Some text and figures contain typos.

- page 4, section title: check the extra "a";

- page 8, check the spelling of "experimentalis";

- all figures: please increase your care for the "spacing" and "layout" of the subpanel titles: they are sometimes wrongly displaced and can get confusing.

- Fig. 1b: check spelling of "Membrane potentia"

- Fig. 2, axes: please use the same symbol for frequency "f" instead of "Fr" and "Z" instead of "imped";

- Fig. 2, use a single convention for indicating the units ("..., MOhm" versus "Phase(rad)")

- Fig. 2a,d: for consistency, use abs(Z) and phase(Z).

- Fig. 2c: the black line for "data1" has not been included

- Fig. 2c: the circuit sketch is "partly occluded" by the legend;

- Fig. 2b,d: use "g_s" instead of "cell G", for consistency with the text;

- Fig. 4c (and 6c): check spelling of "Measured Conductane";

- Fig. 4d,e (and 6d,e): use the same notation "g_e" and "g_i" and not another one as "E" and "I".

- Fig. 4a,d (and 6a,d): check the x-axis label (removing extra space before the closed parenthesis)

Note that the caption of Fig. 3 neither refers nor illustrates panels e.

Note that the caption of Fig. 5 neither refers nor illustrates panels g,h,i,j.

Note that the caption of Fig. 6 contains a typo (i.e. "lowpassong").

5)

- Fig. 2b and d are difficult to grasp quickly;

- Fig. 3f,g,h,i contain almost no information, at the choosen level of representation; consider replacing them by a block diagram, with what is input and what is the filtering/analysis operation.

6) eq. 6: can you use "arctan()" instead of tan^-1 ? It would improve the clarity.

7) Can you explain why you had to use Hilbert and could not use FFT ?

Reviewer #3: Neurons’ activity is by far determined by the interplay between the activity excitatory and inhibitory synapses on their membrane. The effect of an active synapse is to temporarily change the membrane conductance and selectively permit the current of specific ions in or out of the cell. It is thus essential to measure synaptic inputs in neurons to understand how they transform their input into an output. However, previous methods could only allow the measurement of either the excitatory or inhibitory conductance at a time. These methods were helpful in studying the response to a repeating sensory input such that the measurement could be switched from measuring excitation to inhibition. But this could not reveal their co-variation. Neither could it resolve their co-activity when there was no specific stimulus to trigger upon.

This manuscript introduces a novel protocol and analysis method to resolve the excitatory and inhibitory conductances impinging on a neuron simultaneously in a single trial. The study cleverly harnesses the resistance of the measuring electrode (which is often considered a problem/bug). It shows that the combined response of the neuron-electrode system to sinusoidal current injections can be used to resolve the excitatory and inhibitory conductances simultaneously. This is a significant improvement on any existing method and was actually considered an unsolvable problem until now.

The manuscript is very methodological, starting with the simplest case and gradually adds a various level of complexity, solving most of the relevant cases, including variations in the electrode access resistance, electrode capacitance, and synaptic inputs located on a dendritic tree.

I strongly support the publication of the manuscript. I have no major comments on the manuscript. All of my comments relate to clarification of the text. I feel that the paper should take a more careful approach towards the notations. Some of the variables mentioned appear in different equations with different meanings, and this is difficult to keep track of.

General comments:

1. The text on page 8 is unclear. It goes between general statements and specific examples and, in general, is not very methodic. Also, the derivation of equation 8 is not clear at all.

2. In addition, the issue of capacitance compensation is not mentioned at this stage of the text (appears later), while clearly, this is a significant factor as the capacitance of the pipette is much larger than that of the cell. It would be wise to mention here that there is a challenge, and it will be treated later (such that the reader does not think it is ignored).

3. If I understand correctly, equation 5 basically represents the fact that the decay of the impedance is linear with w*C on the log-log plot, as seen in the plot of figure 2a. However, when the electrode is present, this linearity breaks down, as the authors demonstrate on 2c. So is the approximation in eq.5 really justified?

4. The choice of the two fundamental frequencies 210Hz and 315Hz seems somewhat arbitrary. Clearly, if they are too similar there is a separation problem and if one is too high or the other too low there will be interference with the capacitance and resistance. Is there some optimization process that can be suggested to pick the “best” frequencies based on the cell capacitance and the electrode resistance?

5. Equations 11 & 12:

a. The notations of Equations 11 and 12 are not fully clear. The equations use:

i. “c” - which is the estimated capacitance (but the estimation is not used in the symbol)

ii. “Rs” – not clear which Rs is that, the real one, the estimated one from the first procedure (i.e., eq. 6?)

b. So, for example, which is the Rs that appears in equation 12 – is it Rs from equation 11 or is it Rs from equation 6.

c. In addition, in the text right after the equation it refers to Rs_{est2} and G_{est2} in equations 11 and 12 which do not really appear there. These notations should be revised.

d. I might be wrong but if you use delta_w for (w1-w2) it seems that eq. 11 could be simplified.

e. Do the authors have an explanation why G_{est} is only dependent on z1 and w1 and not on z2 and w2? If so, it would be nice to have it in the text following eq. 12.

6. Capacitance compensation: Again, notations are not confusing. The authors define Y=1/Z and then in equation 13 they define Y1 as a different quantity which is no 1/Z so when in eq. 15 they use 1/z1 and Y1 as two different quantities it turns out to be very confusing. This is amplified in the discussion around eq. 17 and 18 where again there is a use in Rs which is used in the definition of Y but also in eq. 11 & 12 (see above).

7. Dendritic input – eq. 19 comes as a surprise with no real justification. It also feels somewhat circular because the estimation of g_s depends on g_l. It would be nice if the authors explain how did they come up with this equation and how do they actually use it.

8. Dendritic inputs – It is well known that the conductance of distal dendritic input cannot be resolved precisely due to space clamp issues. It would be interesting to discuss here the “effective” conductance of the distal inputs. To some extent while the algorithm might not be able to restore the original conductance in the dendrite it might give a good estimate of what this conductance looks like from the soma.

9. In that context consider adding citations to:

a. Häusser M, Roth A. Estimating the time course of the excitatory synaptic conductance in neocortical pyramidal cells using a novel voltage jump method. Journal of Neuroscience. 1997 Oct 15;17(20):7606-25.

b. Rall W, Segev I. Space-clamp problems when voltage clamping branched neurons with intracellular microelectrodes. In Voltage and patch clamping with microelectrodes 1985 (pp. 191-215). Springer, New York, NY.

Minor comments:

The writing could be improved. For example on pp. 3 “Hence, as these methods provide only an average picture and thus fail to capture the instantaneous and trial-by-trial based insight in the relations between excitation and inhibition.”

Figure 1 – there is no legend on the figure showing which color is which.

The pass from page 5 to 6 is not obvious. It would be better to show how you are going to use the impedance in theory and only then add the pipette resistance. Right now the text seems to answer a question that wasn’t even asked at that point.

Legend of figure 2b is sloppy.

Pp6,7 – it would be good to show on a separate panel the intersection point as a function of impedance to support the claim.

P8: Typo: “experimentalis”

P10: “(Figs. 3f to i, due to screen resolution are as patches of colors)” – looks like there is a word missing: “are displayed as patches…”?

Legend of figure 5 could be improved.

Fig 8. It would be better to reverse the order of the labels (i.e., True g_e, measured g_e, VC_g_e)

**********

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Reviewer #2: Yes

Reviewer #3: Yes

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Please note that, as a condition of publication, PLOS' data policy requires that you make available all data used to draw the conclusions outlined in your manuscript. Data must be deposited in an appropriate repository, included within the body of the manuscript, or uploaded as supporting information. This includes all numerical values that were used to generate graphs, histograms etc.. For an example in PLOS Biology see here: http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001908#s5.

Reproducibility:

To enhance the reproducibility of your results, we recommend that you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. Additionally, PLOS ONE offers an option to publish peer-reviewed clinical study protocols. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols

Submitted filename: Comments.docx

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17 Sep 2021

Submitted filename: PLOSCompBio_resubmission_point-by-point001.pdf

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5 Nov 2021

Dear Prof. Lampl,

Thank you very much for submitting your manuscript "A novel theoretical framework for simultaneous measurement of excitatory and inhibitory conductances" for consideration at PLOS Computational Biology. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations.

Regarding the reviews, we believe that the suggestion to include a "case report" is an excellent idea, and would strengthen the impact of the work. Nevertheless, we leave it up to you whether to include that in the present manuscript, and thus submit a new revision, or instead consider a second publication in a different journal to that effect. If you choose not to make the changes, that is fine and you can submit the paper and we will process it as is.

Please prepare and submit your revised manuscript within 30 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email.

When you are ready to resubmit, please upload the following:

[1] A letter containing a detailed list of your responses to all review comments, and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Thank you again for your submission to our journal. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

Lyle J. Graham

Deputy Editor

PLOS Computational Biology

Lyle Graham

Deputy Editor

PLOS Computational Biology

***********************

A link appears below if there are any accompanying review attachments. If you believe any reviews to be missing, please contact ploscompbiol@plos.org immediately:

[LINK]

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: Thanks for your responses. All of my previous major and minor points - except that related to the experimental validation - were addressed. As well, the authors improved the quality of the paper significantly.

I can agree with the statement that " There are

numerous examples from physics, in which theories for measurements were established long

before technologies matured enough to allow full implementation of the method".

However, I want to share this idea with the authors that some detailed discussions related to your recent experience with experimental data (through your new collaboration) can be very useful for the community. It would be great if authors add a "Case Study" in the manuscript and discuss your recent results. Capitalizing on these points in Discussion (I believe you made most of them in your current manuscript) can be significantly helpful for other researchers using/trying your method.

As I am not aware of practical things with respect to this collaboration and the extent to which you can share some data and figures, I leave this idea to authors to check its possibility.

Reviewer #2: Authors have satisfactorily addressed my concerns. I thank them for their effort and I hope we might even be able to collaborate on the actual experimental validation of their interesting novel method.

Reviewer #3: The authors addressed all my comments. I am satisfied with these changes and I think that this revision is much improved. I recommend publishing.

**********

Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data and code should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

**********

PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: Yes: Milad Lankarany

Reviewer #2: Yes: Michele GIUGLIANO

Reviewer #3: No

Figure Files:

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org.

Data Requirements:

Please note that, as a condition of publication, PLOS' data policy requires that you make available all data used to draw the conclusions outlined in your manuscript. Data must be deposited in an appropriate repository, included within the body of the manuscript, or uploaded as supporting information. This includes all numerical values that were used to generate graphs, histograms etc.. For an example in PLOS Biology see here: http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001908#s5.

Reproducibility:

To enhance the reproducibility of your results, we recommend that you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. Additionally, PLOS ONE offers an option to publish peer-reviewed clinical study protocols. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols

References:

Review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript.

If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

30 Nov 2021

Submitted filename: Letter to Editor point by point response_Nov30 - Copy.docx

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6 Dec 2021

Dear Prof. Lampl,

We are pleased to inform you that your manuscript 'A novel theoretical framework for simultaneous measurement of excitatory and inhibitory conductances' has been provisionally accepted for publication in PLOS Computational Biology.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. A member of our team will be in touch with a set of requests.

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.

IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.

Should you, your institution's press office or the journal office choose to press release your paper, you will automatically be opted out of early publication. We ask that you notify us now if you or your institution is planning to press release the article. All press must be co-ordinated with PLOS.

Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology.

Best regards,

Rune W. Berg, PhD

Guest Editor

PLOS Computational Biology

Lyle Graham

Deputy Editor

PLOS Computational Biology

***********************************************************

21 Dec 2021

PCOMPBIOL-D-21-00712R2

A novel theoretical framework for simultaneous measurement of excitatory and inhibitory conductances

Dear Dr Lampl,

I am pleased to inform you that your manuscript has been formally accepted for publication in PLOS Computational Biology. Your manuscript is now with our production department and you will be notified of the publication date in due course.

The corresponding author will soon be receiving a typeset proof for review, to ensure errors have not been introduced during production. Please review the PDF proof of your manuscript carefully, as this is the last chance to correct any errors. Please note that major changes, or those which affect the scientific understanding of the work, will likely cause delays to the publication date of your manuscript.

Soon after your final files are uploaded, unless you have opted out, the early version of your manuscript will be published online. The date of the early version will be your article's publication date. The final article will be published to the same URL, and all versions of the paper will be accessible to readers.

Thank you again for supporting PLOS Computational Biology and open-access publishing. We are looking forward to publishing your work!

With kind regards,

Agnes Pap

PLOS Computational Biology | Carlyle House, Carlyle Road, Cambridge CB4 3DN | United Kingdom ploscompbiol@plos.org | Phone +44 (0) 1223-442824 | ploscompbiol.org | @PLOSCompBiol