Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise

Abstract

A neuron receives input from other neurons via electrical pulses, so-called spikes. The pulse-like nature of the input is frequently neglected in analytical studies; instead, the input is usually approximated to be Gaussian. Recent experimental studies have shown, however, that an assumption underlying this approximation is often not met: Individual presynaptic spikes can have a significant effect on a neuron’s dynamics. It is thus desirable to explicitly account for the pulse-like nature of neural input, i.e. consider neurons driven by a shot noise – a long-standing problem that is mathematically challenging. In this work, we exploit the fact that excitatory shot noise with exponentially distributed weights can be obtained as a limit case of dichotomous noise, a Markovian two-state process. This allows us to obtain novel exact expressions for the stationary voltage density and the moments of the interspike-interval density of general integrate-and-fire neurons driven by such an input. For the special case of leaky integrate-and-fire neurons, we also give expressions for the power spectrum and the linear response to a signal. We verify and illustrate our expressions by comparison to simulations of leaky-, quadratic- and exponential integrate-and-fire neurons.

Appendix A: Expressions for ϕ(v) for various neuron models

Here, we list explicit expressions for

$$ {\phi}(v) = \frac{v}{a} + \tau_{\mathrm{m}} r_{\text{in}} {\int}^{v} dx\; \frac{1}{f(x)} $$

(27)

for PIF, LIF and QIF neurons.

• PIF (f(v) = μ):

  $$ {\phi}(v) = v \left( \frac{1}{a} + \frac{\tau_{\mathrm{m}} r_{\text{in}}}{\mu} \right). $$

  (28)

• LIF (f(v) = μ − v):

  $$ {\phi}(v)= \frac{v}{a} - \tau_{\mathrm{m}} r_{\text{in}} \ln(|\mu-v|). $$

  (29)

• QIF (f(v) = μ + v ²):

  $$ {\phi}(v)\,=\, \frac{v}{a} + \frac{\tau_{\mathrm{m}} r_{\text{in}}}{\sqrt{|\mu|}} \left\{\begin{array}{lll} \arctan\left( \frac{v}{\sqrt{\mu}} \right) & \mu > 0 \\ \!-\text{arctanh}\left( \frac{v}{\sqrt{-\mu}} \right) & \,-\,\sqrt{-\mu} < v \!<\! \sqrt{\,-\,\mu} \\ -\text{arccoth}\left( \frac{v}{\sqrt{-\mu}} \right) & \text{otherwise} \\ \end{array}\right.. $$

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