Relating Neural Dynamics to Neural Coding

We demonstrate that two key theoretical objects used widely in computational neuroscience, the phase-resetting curve (PRC) from dynamics and the spike triggered average (STA) from statistical analysis, are closely related when neurons fire in a nearly regular manner and the stimulus is sufficiently small. We prove that the STA due to injected noisy current is proportional to the derivative of the PRC. We compare these analytic results with numerical calculations for the Hodgkin-Huxley neuron and we apply the method to neurons in the olfactory bulb of mice. This observation allows us to relate the stimulus-response properties of a neuron to its dynamics, bridging the gap between dynamical and information theoretic approaches to understanding brain computations and facilitating the interpretation of changes in channels and other cellular properties as influencing the representation of stimuli.

Figure 1

Estimation of the PRC from the STA for the Hodgkin-Huxley model. (a) STA for two levels of noise (gray: ${\sigma }^2=0.0625{\mathrm{mV}}^2/\mathrm{msec}$; black: ${\sigma }^2=1.0{\mathrm{mV}}^2/\mathrm{msec}$) (b) PRCs reconstructed from the STA (thick line, true PRC; thin lines, reconstructed PRCs).

Figure 2

Estimation of the PRC of a neuron recorded in vitro. Graph shows the STA (black) and the PRC estimated using two different methods from recordings of an olfactory bulb mitral cell. The light gray line shows an estimate from the method described previously [4]. The dark gray line shows average PRC for the same cell calculated from the STA, as described above. (Note the STA has been plotted in time rescaled by $2\pi /T$, the mean period.)

Figure 3

Effect of irregular firing on the estimate of the PRC from the STA for phase models and the HH model. Results of simulations of the phase model with $\Delta (t)=\sin t$ (a1) and $\Delta (t)=1-\cos t$ (a2) with increasing noise amplitude from 0.2 (red)-2.8 (black) in steps of 0.3 (in the order of colors in the rainbow). For both PRCs increasing noise amplitude causes an increase in firing rate. In both cases low noise levels (orange traces) produce most precise estimates of the PRCs. (a3) Reconstruction of the HH PRC with increasing noise values ($\sigma$ shown in figure) (b) Plot of correlation between actual and estimated PRCs vs CV. (Hollow circles, $1-\cos (t)$, solid circles $\sin (t)$ and triangles, HH.)