Do neurons have a voltage or a current threshold for action potential initiation?

The majority of neural network models consider the output of single neurons to be a continuous, positive, and saturating firing rate f (t), while a minority treat neuronal output as a series of delta pulses sigma delta (t-ti). We here argue that the issue of the proper output representation relates to the biophysics of the cells in question and, in particular, to whether initiation of somatic action potentials occurs when a certain threshold voltage or a threshold current is exceeded. We approach this issue using numerical simulations of the electrical behavior of a layer 5 pyramidal cell from cat visual cortex. The dendritic tree is passive while the cell body includes eight voltage- and calcium-dependent membrane conductances. We compute both the steady-state (Istatic(infinity)(Vm)) and the instantaneous (I0(Vm)) I-V relationships and argue that the amplitude of the local maximum in Istatic(infinity)(Vm) corresponds to the current threshold Ith for sustained inputs, while the location of the middle zero-crossing of I0 corresponds to a fixed voltage threshold Vth for rapid inputs. We confirm this using numerical simulations: for "rapid" synaptic inputs, spikes are initiated if the somatic potential exceeds Vth, while for slowly varying input Ith must be exceeded. Due to the presence of the large dendritic tree, no charge threshold Qth exists for physiological input. Introducing the temporal average of the somatic membrane potential <Vm> while the cell is spiking repetitively, allows us to define a dynamic I-V relationship Idynamic(infinity)(<Vm>). We find an exponential relationship between <Vm> and the net current sunk by the somatic membrane during spiking (diode-like behavior). The slope of Idynamic(infinity)(<Vm>) allows us to define a dynamic input conductance and a time constant that characterizes how rapidly the cell changes its output firing frequency in response to a change in its input.