Interpretation of discrete time constants

The purpose of this page is to explain how to interpret the discrete time constants g that appear in the AR formulation of the indicator dynamics. We assume that the recording occurs with a frame rate f Hz, leading to a time bin width of Δt sec. c[ ] denotes the calcium in discrete time, whereas h( ) denotes it in continuous time, so the n-th timestep in continuous time refers to continuous time n*Δt.

Interpretation of AR(1) models

Suppose that a neuron has a discrete time constant g with 0 < g < 1, i.e., c[t] = g*c[t-1] + s[t]. In continuous time this corresponds to an impulse response

h(t) = exp(-t/τ) for t > 0 (and h(t) = 0 for t < 0) where τ is given by τ = - Δt/log(g).

In fact for an input s[0] = 1 and s[t] = 0 otherwise, it holds that c[n] = h(nΔt) for n = 0,1,2,....

Interpretation of AR(2) models

For the case of AR(2) models we consider the model

c[t] = g₁c[t-1] + g₂c[t-2] + s[t]

Let r₁, r₂ be the roots of the polynomial

x² - g₁x - g₂ = 0 with r₁ > r₂

We then compute the continuous time constants τ₁ and τ₂ as

τ₁ = - Δt/log(r₁)

τ₂ = - Δt/log(r₂)

The impulse response in continuous time is then given by

h(t) = exp(-t/τ₁) - exp(-t/τ₂) for t > 0 (and h(t) = 0 for t < 0)

In fact for an input s[1] = 1 and s[t] = 0 otherwise, it holds that

and it holds that c[n]*(r₁ - r₂) = h(nΔt) for n = 0,1,2,....

Note that for AR(2) models in continuous time we use the convention

h(t) = (1 - exp(-t/τ_rise))exp(-t/τ_decay) for t > 0 (and h(t) = 0 for t < 0).

With this convention τ_decay and τ_rise are given by:

τ_decay = τ₁ and τ_rise = (τ₁τ₂)/(τ₁-τ₂)