Firing Rate Noise Assumptions  Linearity of Adapted Current-Discharge

Linear Coupling between Dendrite and Soma

The coupling conductance $G$ and the spiking mechanism define a path for current to flow from the dendritic compartment to ground. We can assign an effective conductance to this current path to be $g_{\text{eff}} = d \langle I \rangle /\langle V_{\text{dend}} \rangle$,where $V_{\text{dend}}$ is the voltage that controls the modulatory voltage-dependent channels in the dendritic compartment. In general, $g_{\text{eff}}$ is itself a function of the mean dendritic voltage $V_{\text{dend}}$. A voltage-dependent effective conductance can have a dramatic effect on the equations governing parameter adaptation, since the full form of these equations includes a factor $${\Bigl[ \frac{\delta}{\delta V_{\text{dend}}(t)}\langle I \rangle \Bigr]}^{-1}.$$ However, when $g_{\text{eff}}$ is approximately constant (i.e., $g_{\text{eff}}$ can be replaced by an Ohmic resistance), then the factor above becomes constant and can be subsumed into the overall learning rate. In fact, this condition can always be achieved by taking the coupling conductance $G$ between the compartments to be much larger than the average conductance from the soma to ground.

Figs. 7 and 8 plot the effective conductance as a function of stimulus amplitude in two conditions, given a coupling conductance $G = 1.5$ $\text{mS}/\text{cm}^2$between the dendritic and somatic compartments. Under the first condition, prior to the neuron ``learning" the appropriate response to stimuli, modulatory calcium and potassium conductances are absent from the dendritic compartment; under the second condition, modulatory conductances are present and have been ``tuned" to their proper settings using eq. 3--this represents the period after learning has been completed.

 

Figure 7: Effective conductance for current to flow from the dendritic compartment through the spiking mechanism to ground. The values on the abscissa represent stimulus amplitudes in the range of stimuli that were likely to occur in the model. (Recall that the stimuli are synaptic conductance inputs drawn randomly from a fixed probability distribution, here a Gaussian probability distribution of mean $\mu = 1.41$ $\text{mS}/\text{cm}^2$and standard deviation $\sigma = 0.25$ $\text{mS}/\text{cm}^2$. The figure shows the effective conductance of the model in its 'native' state, prior to adaptation of the modulatory conductances in the dendritic compartment. For reference, the coupling conductance $G$ and the mean conductance $g_{\text{eff}} = 1.05$are also plotted.

 

 

Figure 8: Effective conductance after the parameters of the potassium and calcium conductances have been adapted to make all firing rates of the neuron in response to the statistical distribution of inputs equally likely. The mean value is $g_{\text{eff}} = 1.04$.

 

Given that the bottleneck to current flow is the coupling conductance and not the spiking mechanism itself, one can replace the effective conductance $g_{\text{eff}}$ by an Ohmic (linear) conductance in the derivation of eq. 3.

By making the two assumptions underlying the particular form of eq. 3, one can write down analytical expressions for the optimal probability distribution of firing rates and the optimal steady-state firing rate as a function of stimulus amplitude. The text defines an auxiliary function $a(f)$ to constrain the firing rates to remain within a fixed range: 

$$a(f) = \begin{cases} f_{\text{min}} \exp \left[ + \frac{... ... f_{\text{max}})}{ \lambda} \right]& f \gt f_h. \end{cases}$$ (vii)

The rule in eq. 3 for adapting the conductance parameters consists of stochastic approximation of the entropy function $S[a(f)] = - \int \ln p[ a(f)] \, p(f) df$.

 With $f( \langle I \rangle) = 1.23 \, ( \langle I \rangle - 7.2)$and $g_{\text{eff}} = 1.04$, the values before learning has occurred, the corresponding parameters to $a(f)$ are:
$$\begin{gather*}\lambda = 1.68\qquad f_{\text{min}} = 24.28 \qquadf_{\text{max}} = 8418. \end{gather*}$$

The extremal solution to $S[ a(f)]$ with $a(f)$ given by eq. vii is
$$\begin{align*}p(f) & = \text{constant} \times \frac{da }{df} \notag \ & = \b... ...{\text{max}}) }{\lambda} \bigr] & f \gt f_{\text{max}}.\end{cases}\end{align*}$$
where $c_1 = \bigl\{ (f_{\text{max}} - f_{\text{min}}) + \lambda + \lambda [1 - \exp (-f_{\text{max}}/\lambda)]\bigr\}^{-1}$.

 Using the shorthand notation $F(x) = \int_{-\infty}^x p(z) \, dz$ for the cumulative distribution and assuming that $f_{\text{max}}/\lambda \gg 0$, one can write the optimal firing rate function as:

$$f(x) =\begin{cases}f_{\text{min}} + \lambda \ln \biggl\{... ...) - F(x) \bigr] + 1 \biggr\} & x_{\text{max}} < x. \end{cases}$$

Since $p(x)$ is Gaussian and hence symmetric about $x = \mu$,we can compute $x_{\text{min}}, x_{\text{max}}$ in terms of $f_{\text{min}}, f_{\text{max}}$ as
$$\begin{align*}x_{\text{min}} & = F^{-1} \biggl[ \frac{c_1 (f_{\text{min}} - ... ...iggl[ \frac{c_1 (f_{\text{max}} - f_{\text{min}}) + 1 }{2}\biggr]\end{align*}$$

The graphs of $f(x)$ and $p(x)$ given by the equations above are shown in the two panels of Fig. 2 in the text, where they represent the optimal steady-state firing rate as a function of the synaptic conductance and the optimal probability distribution of firing rates. 

1/13/1998