^{*}

Analyzed the data: LJC MPB. Wrote the paper: LJC MPB. Conceived and designed the calculations: LJC MPB.

The authors have declared that no competing interests exist.

A recent paper of B. Naundorf

In 1952, Hodgkin and Huxley described the underlying mechanism for the firing of action potentials through which information is propagated in the nervous system. Hodgkin and Huxley's model relies on the opening and closing of channels, selectively allowing ions to move across the membrane. In the original picture, the channels open independently of one another. A recent paper argues that this model is incapable of modeling a set of action potential data recorded in the cortical neurons of cats. Instead the authors suggest that to model their data it is necessary to conclude that ion channels open cooperatively, so that opening one channel increases the chance that another channel opens. We analyze the initiation of action potentials using a method from theoretical physics, the path integral. We demonstrate that deviations of the data from the predictions of the Hodgkin-Huxley model hinge on measurement of the noise strength.

In 1952, Hodgkin and Huxley explained how action potentials are generated through the electrical excitability of neuronal membranes

A recent paper

The result reported in

Here we use a standard technique from theoretical physics (the path integral) to derive an analytical formula relating the onset rapidity and onset span. Our analysis applies to the classical Hodgkin-Huxley model, in addition to generalizations thereof, including those in which the channel opening probability depends on channel density

We first review the essential framework of Hodgkin-Huxley type models for action potential generation. The dynamics of the membrane potential

We are interested in understanding from this model the relationship between onset span and onset rapidity, as defined by

To proceed we use the fact that, at action potential initiation, we need only consider the sodium channels. This is because the potassium channels respond too slowly for their dynamics to influence the voltage

Consider trajectories

Equation (5) is the fundamental equation for the onset span: it requires us to compute

Here the integral is taken over all the possible paths that

It is convenient to rewrite this formula by defining the dimensionless parameters

We have now computed two of the three quantities needed to evaluate Eq. (5) for the onset span

Substituting this into Eq. (6), we obtain

We now can evaluate Eq. 5 for

Numerical evaluation of the function

Asymptotic analysis of the integral in Eq. (12) shows that at small

Thus we have calculated the variance of voltages at which action potential onset occurs as a function of the onset rapidity

Pairs of trajectories simulated using (4) with (A) onset rapidity

To demonstrate the validity of our analysis, we use the reduced Hodgkin Huxley model described by (4) to simulate trajectories and compare the onset span we observe for particular sets of parameter values with that predicted by our analysis. In order to simulate the gaussian noise source

Trajectories (10000) were simulated as in

In both the low

We now compare the theory to the results of Naundorf. In their experiments, they measure the onset span as the difference between the maximum and minimum voltage threshold that is measured. Since 99.7% of observations fall within three standard deviations of the mean, we can approximate the onset span of between 50 and 500 trials as six times the standard deviation

In

Here the solid blue dots are the simulation data points reported in

It is worth noting that additional sources of variance exist when comparing the experiments to the theory. In particular, (i) the theory assumes that the voltage threshold occurs precisely when

The calculations described here clarify that to understand whether the experimental data is consistent with the Hodgkin Huxley picture, it is necessary to understand the corresponding level of

The writing of the paper was done in part at the Kavli Institute for Theoretical Physics.