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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.
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)g = h(nΔt).
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)
and it holds that c(n)*(r1 - r2) = h(nΔt).
Note that for AR(2) models in continuous time we use the convention
h(t) = (1 - exp(-t/τrise))exp(-t/τdecay) for t > 0 (and h(t) = 0 for t < 0).
With this convention τdecay and τrise are given by:
τdecay = τ1 and τrise = (τ1τ2)/(τ1-τ2)