 Constant-input firing rate of a leaky integrate-and-fire neuron

ID: constant-input-firing-rate-of-a-leaky-integrate-and-fire-neuron

For

C_{m} \dot{V} = - g_{L} (V - V_{L}) + I

, reset

V_{0} < V_{θ}

and no explicit refractory interval, define

I_{c} = g_{L} (V_{θ} - V_{L})

and

τ_{m} = C_{m} / g_{L}

. Constant current gives

r (I) = {0, [τ_{m} lo g ((I - g_{L} (V_{0} - V_{L})) / (I - I_{c}))]^{- 1}, I \leq I_{c}, I > I_{c} .

The threshold is approached only asymptotically at

I = I_{c}

. At large current,

r \sim I / [C_{m} (V_{θ} - V_{0})]

. Adding an absolute neuronal refractory period changes the denominator to

T (I) + t_{ref}

and makes the firing rate saturate rather than grow without bound.

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