On the modelling of seizure dynamics.

This scientific commentary refers to ‘On the nature of seizure dynamics’, by Jirsa .

In this issue of Brain, Jirsa and colleagues offer a masterful account of epilepsy that discloses the universal and invariant properties of seizure dynamics (Jirsa ). They derive a formal taxonomy of seizure activity from the basic principles of coupled dynamical systems. In brief, they show that a minimal model—comprising just five states or variables—is sufficient to describe the onset, time-course and offset of seizure activity. This may sound implausible; however, the repertoire of dynamics that coupled systems can generate may be much smaller than people imagine. In effect, Jirsa et al. use their understanding of dynamical systems to ‘diagnose’ the mathematical aetiology of the ‘signs and symptoms’ expressed in the statistics of seizure activity. Put simply, the existence of dissociable fast discharges and spike-wave events immediately tells them that the dynamics must be generated by two pairs of (hidden) coupled states generating fast and spike-wave oscillations, respectively. The fact that seizure activity waxes and wanes over a slower timescale then calls for a further slow permittivity variable that controls the expression of fast dynamics. Within this model, the onset and offset of seizure activity become well-defined mathematical objects: a saddle node and homoclinic bifurcation, respectively. The tell-tale signatures of these bifurcations were subsequently confirmed using in vitro experiments and focal seizures recorded in humans and zebrafish. So why is this important?

Without recourse to Wikipedia, I confess that I could not explain the difference between a saddle node and homoclinic bifurcation. However, this is not important: the authors have done all the mathematical heavy lifting for us and bring three things to the table: first, the notion that seizure onset (and offset) can be cast in terms of bifurcations (qualitative changes in dynamics) that are induced by changing the parameters of a coupled dynamical system—such as synaptic efficacy in neuronal microcircuits. Second, the attending qualitative changes rest upon a separation of temporal scales. For example, slow fluctuations in synaptic efficacy induce qualitative changes in fast dynamics. Finally, there is an inherent circular causality in epileptogenesis; whereby slow fluctuations in synaptic efficacy cause changes in fast neuronal activity—that couple back to the slow variables through processes such as activity-dependent plasticity. This circular causality is a necessary aspect of coupling variables that fluctuate over separable timescales. It is interesting to note that these fundaments of dynamical systems are formally articulated in synergetics—an approach to self-organization pioneered by the first author’s mentor Herman Haken (Haken, 1983). In short, what is on offer here is a canonical description of seizure activity that has all the necessary ingredients for a formal taxonomy of seizure dynamics. But how does a mathematical taxonomy reach beyond phenomenology?

At this point, we have to consider why the modelling of seizure dynamics is potentially important. In previous years, there have been substantial advances in the use of models of neuronal microcircuits as forward or generative models of observed electrophysiological responses. This is most evident in the advent of dynamic causal modelling, which is now used routinely to address questions about functional brain architectures—and their synaptic underpinnings—in cognitive and systems neuroscience. The key thing here is that dynamic causal models are not just used to reproduce the phenomenology of electrophysiological measurements: they explain empirical data through Bayesian model inversion and subsequent selection. A formal understanding of how seizure dynamics are generated could therefore provide vital constraints on dynamic causal models of seizure activity. In principle, this means that one could identify the biophysical substrates of slow and fast variables in phenomenological models.

A brief history of epilepsy models

The generative mechanisms underlying seizure activity have been modelled extensively; particularly in neocortical and hippocampal systems. A recurring theme in these studies is the interaction between excitatory pyramidal cells and inhibitory interneurons. This speaks to the central role of intrinsic connectivity in maintaining excitation-inhibition balance. The intracellular and extracellular mechanisms underlying slow changes in synaptic efficacy have been the subject of much study. For example, Wendling model seizure onset in terms of slow ensemble dynamics involving pyramidal cells and local interneurons, highlighting the increases in excitability and decreases in slow dendritic and fast synaptic inhibition that peak at seizure onset. The emerging picture from these studies points to slow changes in the excitability of cortical and subcortical microcircuits mediated by interactions between pyramidal cells and inhibitory interneurons. Slow fluctuations—and associated extracellular increases in potassium and decreases in calcium ion concentration—cause, and are caused by, the fast fluctuations in synaptic activity that announce seizure onset. Indeed, Nevado-Holgado have characterized the evolution of seizure activity as a path through the parameter space of a neural mass model, whereas Hocepied consider a similar formulation for seizure detection. But how can we test these hypotheses—or indeed quantify synaptic variables in a given patient at a given point in time? Here, we turn to dynamic causal modelling.

Dynamic causal modelling of seizure dynamics

Dynamic causal modelling (DCM) is a biophysically informed framework for comparing hypotheses or network models of (neurophysiological) time series. It is an established procedure in the analysis of functional magnetic resonance time series and is now used increasingly in electrophysiology. DCM rests upon the fact that there is a straightforward mapping between the synaptic connectivity of neuronal circuits and their responses to exogenous or endogenous neuronal input. In some cases, this mapping can be remarkably direct and intuitive (see Fig. 1 for an example).

Figure 1

This schematic illustrates the mapping between the (synaptic efficacy) parameters of a dynamic causal model and their spectral signatures. Left: A simple dynamic causal model of recurrently and reciprocally coupled excitatory (black) and inhibitory (red) neuronal populations. Middle: The corresponding spectral responses under linear coupling. The equation (based on standard dynamical systems theory) shows that the (Lorentzian) spectral density function of frequency (ω) is centred on the connection strength of reciprocal connections (β), while the dispersion (full width at half maximum) of the spectral peak is determined by recurrent connection strengths (α). Connection strengths are naturally converted into frequencies because (in dynamical models) connections are measured in hertz—and therefore play the role of rate constants. Right: Time-dependent changes in the spectral peak therefore reflect changes in the strength of intrinsic (reciprocal) connectivity.

There is an extensive literature on DCM ranging from face validation studies (David ) to construct validation in terms of multimodal measurements and pharmacological manipulations (Moran ). Crucially, DCM is now finding a role in understanding pathophysiology in terms of extrinsic (long range) disconnections (Boly ) and aberrant intrinsic (local) connectivity that may underlie pathological (beta) oscillations in Parkinson’s disease (Marreiros ). The hope here is that DCM can be used to quantify slow fluctuations in synaptic efficacy in epilepsy—and perhaps to predict seizure onset. The promise for modelling seizure dynamics is illustrated in Fig. 2. This figure shows the spectral responses of a canonical microcircuit as a function of intrinsic connectivity; namely, the recurrent self-inhibition of superficial pyramidal cells. This simple manipulation leads to fast epileptiform activity as the system approaches a (transcritical) bifurcation—and is motivated easily by the convergent evidence implicating fast synaptic inhibition in the context of recurrent coupling between pyramidal cells and inhibitory interneurons (see above).

Figure 2

(A) A simple two source architecture with the (four) populations of a single source highlighted. The intrinsic connectivity (dotted lines) and extrinsic connectivity (solid lines) conform to a canonical microcircuit and the known laminar specificity of extrinsic connections. Excitatory connections are in red and inhibitory connections are in black. Endogenous fluctuations drive the input cells and measurements are based on the depolarization of superficial pyramidal cells. (B) These summarize the response characteristics of a single source in terms of the impulse response in frequency space, shown graphically (left) and in image format (right). These are (modulation) transfer functions of (the log scaling of) recurrent inhibitory connectivity (solid red line in the microcircuit). (C) These panels illustrate changes in neuronal activity when increasing recurrent inhibition. The top panel shows the strength of recurrent inhibition as a function of time, while the second panel shows a simulated response obtained by integrating the neural mass model on the left with random fluctuating inputs. The empirical time frequency is shown below in terms of the spectral power over 1 to 128 Hz. The lower panel shows the predicted response based upon the transfer functions in B. LFP = local field potential.

The power of DCM lies in its ability to adjudicate among different mechanistic hypotheses using Bayesian model selection. For example, David analysed seizure activity elicited during electrical stimulation. Bayesian model comparison suggested that seizure onset could be modelled in terms of changes in intrinsic connectivity (as opposed to extrinsic connectivity). Furthermore, DCM identified fluctuations in synaptic efficacy in the ictal zone; suggesting that the mesial temporal lobe epilepsy studied by David could be explained in terms of a pre-ictal increase in sensitivity to hippocampal afferents from the temporal pole.

Conclusion

This sort of study provides proof-of-principle that the functional anatomy of epilepsy can be characterized in single subjects. If Jirsa et al. are correct; the universal anatomy of seizure dynamics—and associated neural mass models—suggests that we are in a position to create universal models of seizure activity. However, there is one outstanding challenge: a key insight afforded by Jirsa et al. is the circular causality of seizure dynamics. Currently, dynamic causal models only consider the enslaving of fast variables by slow variables but not vice versa. The approach of Jirsa et al. may provide formal constraints on how fast synaptic activity couples back to slow changes in synaptic efficacy; in other words, the induction of activity-dependent plasticity by fast dynamics. One hopes that modelling this key link will attract further attention over the years to come.

Funding

K.F. is funded by the Wellcome Trust. Principal Research Fellowship (Ref: 088130/Z/09/Z).