Learning Rule  Dendritic Compartment

Equations

For a fixed synaptic conductance $g_{\text{syn}}$,the evolution of the voltage in the dendritic compartment is described by: $$\begin{split}C \frac{d V_{\text{dend}}}{dt} = & g_{\text{s... ... \bigr)\ & - G (V_{\text{dend}} - V_{\text{soma}}).\end{split}$$

Here $g_{\text{syn}}$ is drawn from a Gaussian distribution of mean $\mu = 1.41$ $\text{mS}/{\text{cm}}^2$and standard deviation $\sigma = 0.25$. The noise $\xi(t)$  also has standard deviation $\sigma = 0.25$. The synaptic reversal potential, which limits the stimulus-driven current entering the dendritic compartment, is set to $E_{\text{syn}} = 5 \; \text{mV}$. The reversal potential for $\text{Ca}^{2+}$ is $E_{\text{Ca}} = 70 \; \text{mV}$, similar to that of $\text{Na}^+$, while the potassium reversal potential remains $E_{\text{K}} = -72\; \text{mV}$ mV, as in the somatic compartment. Both somatic and dendritic compartments have the same resting membrane conductance $g_{\text{memb}} = 0.3 \; \text{mS}/\text{cm}^2$, conforming to the original Hodgkin-Huxley model.
    To generate the noise $\xi(t)$, pseudo-random Gaussian variables in frequency space are matched to a given power spectrum for the noise--in our case, the spectrum is chosen to be uniform (flat) up to a cut-off frequency of 500 Hz; an inverse Fourier transform yields noise waveforms of 131072 millisecond duration, sampled at one millisecond intervals. The numerical routines for solving the Hodgkin-Huxley differential equations linearly interpolate between the sampled values of the noise waveform.

The activation and inactivation variables $m_i$ and $h_i$obey
$$\begin{align*}\tau \frac{d m_i}{dt} & = m_{\infty,i}[V_{\text{dend}}(t)] - m_i\... ..._\infty(V) & = 1/\{1 + \exp[ - (V- V_{\frac{1}{2},h_i}) /s_{h_i}]\}\end{align*}$$

All peak conductances for the modulatory potassium and calcium conductances were initially set to zero. The midpoint voltages for $\text{K}^+$ and $\text{Ca}^{2+}$ conductances were spaced evenly between $-60$ and $10 \; \text{mV}$, such that the $\text{K}^+$ conductances occupied the low end of the voltage range. Midpoint voltages for inactivation were uniformly offset by $20 \; \text{mV}$from the activation midpoints. The slope for all activation functions was initially set to $10 \; \text{mV}$, for inactivation functions, $-10 \; \text{mV}$. The initial conditions are summarized in tables 1 and 2.

peak conductance act. midpoint act. slope inact. midpoint inact. slope

$$\bar{g}_i/g_{\text{leak}} V_{1/2} s V_{1/2} s$$

0.0 -20.0 10.0 0.0 -10.0

0.0 -10.0 10.0 10.0 -10.0

0.0 0.0 10.0 20.0 -10.0

 
 
 

peak conductance act. midpoint act. slope inact. midpoint inact. slope

$$\bar{g}_i/g_{\text{leak}} V_{1/2} s V_{1/2} s$$

0.0 -60.0 10.0 -40.0 -10.0

0.0 -50.0 10.0 -30.0 -10.0

0.0 -40.0 10.0 -20.0 -10.0

 

These parameters are modified as the neuron model learns the statistics of the inputs. The equations that govern parameter adaptation are described in detail in the next sections. The adapted parameter values for the curves labeled 'learned' in Fig. 2 of the text are listed under adapted parameters.

1/14/1998