We will consider here networks of stochastic leaky integrate-and-fire neurons, that evolve in discrete time according to:

v_{i}(t) = v_{i}(t − δt) exp(−δt/τ_{i}) + ∑_{j} w_{ij}(t − δt)f_{j}(t − δt)

The neuron fires stochastically with probability σ(v_{i}(t)) If the neuron fires (f_{i}(t) = 1), the potential is reset to a base value (reset potential),

σ_{i}(v_{i}) = \{ \begin{matrix}δt/τ_{σ} exp(β_{σ}(v_{i} − θ_{i})) & if less than 1 \\ 1 & otherwise \end{matrix}

The neurons had a time constant τ_{i} = 20ms, reset potential V_{r} = 10 mV, threshold potential θ_{i} = 16 mV, exponential escape noise with τ_{σ} = 20 ms and β_{σ} = 0.2 mV.

The synaptic weights were initialized with random values at the beginning of the experiment.

The activation of the input neuron was proportional to ... The activations were normalized between 0 and 1. The input neurons fired Poisson spike trains, with a firing rate proportional to the activation, between 0 and 50 Hz.

P (one spike during δt) ≈ rδt

This equation can be used to generate a Poisson spike train by first subdividing time into a bunch of short intervals, each of duration δt. Then generate a sequence of random numbers x[i], uniformly distributed between 0 and 1. For each interval, if x(i) < rδt, generate a spike. Otherwise, no spike is generated. This procedure is appropriate only when δt is very small, i.e, only when rδt ≪ 1. Typically, δt = 1 msec should suffice. (e.g. r = 100 spikes/second).

The spikes of motor neurons were converted to effector activations by integrating them with a leaky accumulator of time constant τ_{e} = 2s. The motor activations a evolved according to

a(t) = a(t − δt) exp(−δt/τ_{e}) + (1 − exp(−1/ν_{e}τ_{e}))f(t)

where f(t) indicates whether the motor neuron has fired. The factor that weighed the contribution f(t) of the spikes insured that the activation was normalized to 1 when the neuron fired regularly with frequency ν_{e} = 25 Hz; the activation was also limited to the interval [0, 1] through hard bounds. The activations of 2 motor neurons were averaged to yield the activation of one effector.