Linear Coupling between Dendrite  Assumptions

Linearity of Adapted Current-Discharge Relationship

  In response to current injection, the HHA model undergoes a transition to spiking behavior typified by the following current-discharge relationship in the absence of adaptation: $$f(I ) = c \sqrt{I - I_\theta},$$ where $c$ is a constant and $I_\theta$ is a threshold current. This functional behavior is a universal property of an entire class of Hodgkin-Huxley models, of which the HHA model is but one example. The class of such spiking models is characterized by arbitrarily low firing rates near threshold and can be analyzed by the classical mathematical techniques of local bifurcation analysis. For instance, the analysis of linear stability predicts that the threshold current for firing an action potential in the HHA model is $I_\theta = 8.11127$ $\mu\text{A}/\text{cm}^2$.

 Since firing rate adaptation occurs on a timescale that is long compared to the typical interspike interval, we can deduce the current-discharge relationship for the adapted neuron from the ansatz: 

$$f(I) = c \sqrt{I - I_\theta - I_{\text{adapt}}},$$ (v)

in which we have assumed that the adaptation current $I_{\text{adapt}}$ is approximately constant throughout the firing cycle. The self-consistent solution to eq. v is 

$$f(\langle I \rangle) = (\langle I \rangle - I_\theta)/k$$ (vi)

for currents near threshold, where $k$ is the linear constant of proportionality in the power series expansion of the adaptation current in terms of the firing frequency, $I_{\text{adapt}} = k_0 + k f + h f^2 + \dots$.

 According to eq. vi, adaptation linearizes the firing rate as a function of input current. This linearization effect is a general property of adaptation for nearly any current-discharge relationship. In Fig. 5 we compare the firing rate response of the HHA model with and without the adaptation current used for the model.

 

Figure 5: Steady-state current-discharge relationships of the HHA model in the presence or absence of firing rate adaptation. A single electrotonic compartment's response to constant current injection is plotted. The linear fit to the adapted curve  (blue dashed line) in the units shown below is $f(I) = 1.15 (I - 3.17)$.

 

The measured current-discharge relationships of mammalian cortical neurons in experimental current injection studies, both in slice and in anaesthetized animals, tend to be linear or nearly linear (Granit et al (1963); Mason and Larkman (1990); Jagadeesh et al. (1992), Ahmed et al. (1995)) .

Figure 6 plots the average firing rate of the full model with modulatory conductances in the dendritic compartment as a function of the average current reaching the soma between spikes. To match the conditions under which learning occurs, firing rates are measured over the entire stimulus interval, and thus include the transient behavior of the Hodgkin-Huxley equations to the onset of stimuli. The noise in the firing rate has been averaged out by repeated trials.  In general, there is little difference in this current-discharge relationship before and after adaptation.

 

Figure 6: Current-discharge relationships in the two-compartment model before and after the model has ``learned" the optimal representation of conductance inputs by adjusting the modulatory potassium and calcium conductances in the dendritic compartment. In this case, the neuron has adapted itself to use all firing rates equally often. $\langle I\rangle$ is the mean current that flows from the dendritic to the somatic compartment during the interspike interval. The dot-dashed line is the best linear fit to the current-discharge relationship before learning: $f( \langle I \rangle) = 1.17 \, ( \langle I \rangle - 4.61)$.

 

Before parameter adaptation, the best linear fit to the current-discharge relationship is $f( \langle I \rangle) = 1.17 \, ( \langle I \rangle - 4.61)$; after parameter adaptation, the current-discharge relationship changes slightly to $f( \langle I \rangle) = 1.23 \, ( \langle I \rangle - 7.2)$.

1/13/1998