The Biophysical Basis of Firing Variability in Cortical Neurons - Computational Neuroscience: A Comprehensive Approach - page 166

To remove the latter effect, we can instead deliver smoothly decaying ramps of cur-
rent. The variance of the time of occurrence of spike i is determined as:

1
N i

2

=

i (

t i )

v i t ij − (6.2)

j , n j ≥

where t ij is the time of spike i in response j , N i is the number of responses with i or
more spikes, n j is the total number of spikes in response j , and t i is the mean time
of spike i . The slope of this quantity with time is the rate of generation of spike time
variance.
Figure 6.5A-C shows, in several different cells, that the accumulating variance
goes through two phases as the stimulus decays, a low-variance stage, and a high-
variance stage. The variance in the late stage begins to approach that of a Poisson
process - that is the increment in variance per spike approaches the square of the
mean spike interval. In contrast, with a steady plateau stimulus, there is, apart from
slight initial adaptation, a steady rate of variance generation.

6.6 Noisy spike generation dynamics
What is the biophysical basis of these two stages of variance generation? Although
we know that there are many different voltage-gated channels which activate and
deactivate over different timescales in cortical neurons [42], many of these are at
much slower time scales than the spike. Initiation of axonally-propagated spikes is
dominated by populations of fast Na channels and fast K channels, in a restricted
site in the neuron, around the soma and initial segment of the axon [57]. The qual-
itative features of this two-stage behaviour are displayed by a model as simple as
an isopotential patch of membrane with stochastically-simulated Na and K channels
whose states and transition rates are derived from the deterministic Hodgkin-Huxley
model (for details of the model, see [9]). In this model, voltage-dependent proba-
bilistic transitions between channel states are simulated explicitly, and the level of
voltage noise increases as the area of membrane (and therefore number of channels)
is reduced. At four different membrane areas, the two stages of rising spike time
variance are clearly seen in response to the same decaying ramp of current density
( Figure 6.6 ).
An essential difference between the two stages of rising variance is now seen:
the gradient of the early stage is highly sensitive to the noise level, increasing in
inverse proportion to membrane area, while that of the late, high variance stage is
almost independent of membrane area. Mean firing frequency decays only slightly
during the burst. The point of transition between low and high variance stages is also
sensitive to the noise level. Perhaps surprisingly, the higher the membrane area (i.e.
the lower the noise level), the earlier the transition. This effect leads to a crossover
of the relationships at about 270 ms (indicated by an arrow).