# Renewal process and perfect integrate-and-fire model with random threshold

The family of integrate-and-fire (IF, aka IAF) models is a popular class I point neuron modeling. A typical IF neuron integrates the input current, and when the membrane potential reaches the fixed threshold it generates an action potential and resets its membrane potential. Since membrane potential is the only dynamic variable, each inter-spike interval (ISI) is independent from each other. Hence IF neurons are generators for renewal processes. This link from IF to renewal process is not one-to-one. There could be many models that generate the same renewal process. In some special cases with stationary input and noise statistics, the analytical solution for the inter-spike interval distribution of an IF model can be obtained through solving the first passage time density [1].

If we consider perfect integrate-and-fire (PIF) neuron with random threshold, a stationary renewal process can be easily mapped in a unique manner. A perfect integrator can be represented as . We normalize the scale such that the phase like variable represents time, and the integration time becomes . When reaches the threshold, an action potential generation is marked and is reset to 0. Without any randomness, this is the simplest oscillator. Now, let be the random variable for the inter-spike interval of a stationary random process of interest. One can think of the process as a noisy oscillator. If we reset the threshold of the PIF neuron to be drawn from each time it fires, it is easy to show that the PIF behaves exactly as the renewal process. Hence, the PIF with random threshold and stationary renewal process are one-to-one mapped (inspired by [2]).

A major advantage of PIF model compared to the renewal model is that we can implement (potentially non-stationary) input dependence naturally. Since the behaves as the phase variable of a noisy oscillator, we can use the phase response curve (PRC) formalism. A PRC is defined as the phase change to a infinitesimal perturbation of the oscillator. Therefore the PIF with small input can be well approximated with, (e.g. [3]).

This model has two parameters to estimate: and . can be estimated from ISI distribution of spontaneous activity. Once we have , can be estimated by the distortion of the firing time distribution given small perturbation at different phases. However, for large , it is difficult to estimate this quantity, simply because of the rarity of events with longer interval in general. Hence, a parametric approach is preferable for estimating the PRC.

References:

- Lindner’s lecture 2009
- Lars Wolff, Benjamin Lindner. Method to calculate the moments of the membrane voltage in a model neuron driven by multiplicative filtered shot noise. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 77, No. 4. (2008), 041913.doi:10.1103/PhysRevE.77.041913
- Roberto F. Galan, Nicolas Fourcaud-Trocme, G. Bard Ermentrout, Nathaniel N. Urban. Correlation-Induced Synchronization of Oscillations in Olfactory Bulb Neurons. J. Neurosci., Vol. 26, No. 14. (5 April 2006), pp. 3646-3655. doi:10.1523/JNEUROSCI.4605-05.2006