One and Two Compartment Stochastic Integrate-and-Fire Neural Models
by
Charles E. Smith
Biomathematics Program, Dept. of Statistics, North Carolina State Univ
Coauthors: Mamiko Arai, Biomathematics Program, N C State Univ.

One and two compartment stochastic integrate-and-fire neural models are investigated by simulation and by analytic approximation methods. The models used are motivated primarily by the papers of Lansky and Rodriguez (1999 a, b). One main difference is that the output of our model is a renewal process rather than a correlated point process. Biophysically this corresponds to antidromic invasion of the action potential into the dendrite to reset the membrane voltage following an action potential.

We concentrate on two neurons, both with the same compartment(membrane electrical properties) at the site of action potential initiation, however the two compartment model includes the dendritic partition of the neuron by a second compartment. The two compartment model is to contrast spatial effects in neurons with longer thinner dendrites to those with short thicker dendrites that can be modeled as a one compartment equivalent circuit. Biophysically this means that the voltage is roughly the same in the spike initiation site and the proximal dendritic processes.

Euler forward method is used to simulate the Ito version of the stochastic differential equations corresponding to these equivalent circuits. A Wiener process is used as the noise term to represent many smaller synaptic inputs and paralleling the approach of Lansky and Rodriguez. The approximation methods for first passage times outlined in Smith(1991) were computed and compared to moments of simulated output of the neurons.

The shapes of the simulated histograms were fit by a normal, gamma and third and fourth order Laguerre series approximations using the method of moments. The corresponding fits and moment plots (skew vs CV; excess vs. skewness (Pearson plot)) were examined. The single gamma seemed adequate in most cases.

The simulation was done in a 3 factor experimental design and blocked on type of model (one vs. two compartment). The factors were: (1) value of voltage threshold for firing “ S “; (2) synaptic input strength “u “; and (3) intensity of the noise input “k”.

For smaller noise variation and when the mean voltage crosses the threshold a heuristic explanation explains the systematic variation in the mean and standard deviation of the first passage time, namely the mean interval of the output point process and its standard deviation. The approximation is simply that expected from the delta method. The standard deviation of the firing time is approximately the standard deviation of the voltage divided by the slope the mean voltage trajectory. Both voltage terms are evaluated at time t* which is when the mean voltage trajectory reaches the threshold S.

For equal firing rates or output mean intervals, the two compartment model shows a pronounced reduction of variability in firing times. Said differently, it can more effectively code input intensity levels using a mean rate neural coding scheme since it has less variability about the mean firing time.

Finally some suggestions for further work are presented.

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Date received: April 4, 2008