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\noindent
We will consider here networks of stochastic leaky integrate-and-fire neurons, that evolve in discrete time according to:
\begin{displaymath}
v_i(t)= v_i(t - \delta t)\exp (-\delta t/\tau_i) + \sum_j w_{ij}(t - \delta t)f_j(t - \delta t)
\end{displaymath}


\noindent
The neuron fires stochastically with probability $\sigma(v_i (t))$
If the neuron fires ($f_i(t)$ = 1), the potential is reset to a base value (reset potential),


\begin{eqnarray*}
\sigma_{i}(v_i) = \left\{
\begin{array}{ll}\nonumber
\delta t/\tau_{\sigma} \exp(\beta_{\sigma}(v_i - \theta_i))& \hspace{.2in}\mbox{if less than 1} \\
1                                                          & \hspace{.2in}\mbox{otherwise}      \\
\end{array}
\right.\\
\end{eqnarray*}

\noindent
The neurons had a time constant $\tau_i$ = 20ms, reset potential $V_r$ = 10 mV, threshold potential $\theta_i$ = 16 mV, exponential escape noise with $\tau_{\sigma}$ = 20 ms and $\beta_{\sigma}$ = 0.2 mV.\\

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

\noindent
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.
\begin{displaymath}
P(\mbox{one spike during} \hspace{.07in} \delta t) \approx r \delta t
\end{displaymath}
This equation can be used to generate a Poisson spike train by first subdividing time into a bunch of short intervals, each of duration $\delta t$. Then generate a sequence of random numbers $x[i]$, uniformly distributed between 0 and 1. For each interval, if $x(i) < r \delta t$, generate a spike. Otherwise, no spike is generated. This procedure is appropriate only when $\delta t$ is very small, i.e, only when $r \delta t \ll 1$. Typically, $\delta t = 1$ msec should suffice. (e.g. $r$ = 100 spikes/second).\\

\noindent
The spikes of motor neurons were converted to effector activations by integrating them with a leaky accumulator of time constant $\tau_e$ = 2s.
The motor activations $a$ evolved according to
\begin{displaymath}
a(t) = a(t - \delta t)\exp(- \delta t/\tau_e) +(1-\exp (-1/\nu_e \tau_e))f(t)
\end{displaymath}
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 $\nu_e$ = 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.


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