Neuronal interspike time histograms for a random input model.

It is currently accepted that the randomness of the interspike intervals of neurons may originate from either variability in the synaptic input or the spike production process or both. In a recent article (Levine and Shefner, 1977) an interesting model for retinal ganglion cells of goldfish was proposed and analyzed, in which the input was fixed and all the variability arose from the spike production process. The experimental histograms of intervals were found to be well fitted to "hyperbolic normal" densities or linear combinations thereof. I wish to comment on the results obtained when the randomness is contained only in the synaptic inputs. The neuron under consideration has a trigger zone with depolarization V(t). The model employed is a slight modification of the "leaky integrator" model, in that excitatory and inhibitory postsynaptic potential amplitudes depend on how close the membrane potential is to the corresponding reversal potentials. Let s be the time constant of decay: the model is summarized by the stochastic differential equation dV(t) =-(V(t)/s)dt + g9(VE-V(t))dP(fE;t) + g1(V1-V(t))dP(f1;t), with V(O) = 0. Excitatory and inhibitory inputs occur according to events in the independent Poisson processes denoted P(fE;t) and P(f1;t) where fE and fi are the corresponding mean rates. The reversal potentials are VE and VI and gE, g, are constants. The quantity of interest is the random variable T, which is the time at which V(t) first reaches or exceeds the threshold 0. The density of Tis very difficult to find analytically for this model, though sometimes expressions can be found for the first few moments (Tuckwell, 1976). Hence computer simulations were performed for various amounts of excitatory and inhibitory inputs. Results are shown in Fig. 1 for two cases, where histograms of interspike intervals are plotted with time on a logarithmic scale. The parameters were s = 5.8 ms, a typical value for central nervous system 300 / 200 f\ EX ONLY n(t):1 log,0 tpm s FIGURE I Histograms, with time plotted logarithmically, of interspike intervals obtained with the model neuron described above. The squares are for excitation only, triangles for excitation with inhibition.