Biophysical Substrate of Adaptation  Assumptions  Linear Coupling between Dendrite

Noise in the Firing Rate

The criterion for optimizing the neuron's behavior is the following lower bound (see text) on the mutual information between the firing rate $f$ and the stimulus $x$:

$${\cal{I}}_{\text{LB}} (f;x) = - \int \ln \,\biggl( p(f) \, ......ma_f(x) \biggr) \, p(x) \, dx- \ln (\sqrt{2 \pi e}),\tag{2}$$ (2)

 

This quantity depends on $\sigma_f$, the standard deviation of the firing rate, which is a measure of the noise in the model neuron's output.

The simplifying assumption made in the derivation of the learning rule for adaptation (eq. 3 of the text) was that $\sigma_f$ is constant. In this section, we will show how well this assumption is satisfied in practice.

Note that the standard deviation of the firing rate is not solely, or even primarily, determined by the synaptic conductance noise, since two other sources of variance contribute, namely the discrete nature of spikes and the system's dynamical time constants. To see why, recall that firing rates are measured by the number of spikes in finite time intervals of a fixed length $T$ = 200 msec. Even if the model neuron were deterministic and exhibited no transient behavior, the integer number of spikes measured within an interval would still depend on the neuron's phase relative to the first spike at the start of the measuring interval. Furthermore, the combined presence of many neuronal time constants--belonging to the adaptation, calcium, and potassium currents--will make the spike count in the current interval $T$ depend on the stimulus history in previous intervals. So, even in the completely deterministic case, the firing rate is not simply a function of the stimulus at that instant.
 
 

 

Figure 9: Standard deviation of the firing rate, plotted against the mean synaptic conductance, before and after the model neuron's modulatory conductances have been adapted to produce a uniform distribution of spike counts. Compare with the mean firing rate plotted against synaptic conductance shown in Fig.  2a of the text.

  For the  model neuron without any modulatory conductances (before parameter adaptation),  the integer nature of the spike count leads to an aqueduct-arc-like structure of the graph curve of the standard deviation.  This arc-like modulation of this curve around its mean value is, however, not pronounced. In addition, note that the standard deviation of the firing rate is typically one order of magnitude less than the typical firing rate.
   
 Biophysical Substrate of Adaptation  Assumptions  Linear Coupling between Dendrite
1998-08-16