Table 1.
Symbol nomenclature in the order of appearance.
Figure 1.
Encoding in the mean and variance channel.
(A) Simultaneous increase of excitatory and reduction of inhibitory activity (or vice versa) results in a mean current change (right, green). On the other hand, simultaneous increase (or reduction) in excitatory and inhibitory spiking activity results in modifications of the net current variance (left, red). These modifications constitute two primary channels of communication in a cortical network. (B) In a cortical network the excitatory and inhibitory currents add up such that the net somatic current is only weakly correlated across neurons [31], [32].
Figure 2.
Computational role of mean-encoded signals.
(Top) Representation of periodic mean stimuli in the population rate of noisy, independent neurons. (Left) Representation of step-like mean signals in the population rate of noisy, independent neurons. (Bottom) Common fluctuating currents from presynaptic partners represent a common mean signal that leads to pairwise spike correlation function . (Right) The average voltage before a spike is shaped by the linear mean response.
denotes the input current correlation function. The role of the linear response function
is indicated by a dashed green line. Results obtained in the alternative threshold model are discussed in the indicated Sections of this manuscript.
Figure 3.
Spike generation and signal representation in the single spiking threshold neuron.
(A) Spike generation from a temporally correlated Gaussian voltage trace in a single threshold neuron. (B) Encoding of common signals by the population firing rate of independent threshold neurons. Note, that
can be either linearly related to the stimulus (linear regime for weak signals, Eq. 11, 12) or be described by a non-linear response function (e.g. see Eq. 20).
Figure 4.
Linear response to mean and variance modulations in a population of independent threshold neurons.
(A) Normalized amplitude vs.
in response to mean current modulations, simulations (circles) and analytical results in Eq. 21 (solid line). (B)
vs.
in response to current variance modulations, simulations (circles) and analytical results in Eq. 24 (solid lines). Regimes of high-pass and low-pass behavior for linear response function for mean (C) and variance modulations (D). Note, vector strength
in (A) and (B) is proportional to the linear response
, see Eq. 53.
Figure 5.
Firing rate response to a step-like current signal at time
in a population of independent threshold neurons. (A,B) Firing rate change
in response to a mean current step-like increase of amplitude
, amplitude-to-threshold ratio
. Analytical solution in Eq. 25 (solid lines) and simulation results (circles) are superimposed. (A)
for
,
and varying current correlation times
. (B)
for
,
and
. (C,D) Firing rate change
in response to an step-like increase of the current variance
. (C)
and
,
. (D)
for
,
and
. Analytical solution in Eq. 26 (solid lines) and simulation results (circles) are superimposed. Note that the evoked change of the stationary firing rate in A and C, B and D is the same.
Figure 6.
Fidelity of the linear approximation in relation to the complete non-linear response.
(A) Schematic illustration of how well the linear approximation of the population rate derived for low amplitudes (as in Eqs. 25,26) captures the complete response dynamics. (B) Differences in linear and non-linear population firing rate in response to mean current steps of different amplitudes. Here, the linear response corresponds to Eq. 25 and the non-linear response derives from Eq. 20; stationary firing rate and
. (C) Differences between the complete population firing rate (solid line, Eq. 20) and its linear approximation (dashed line, Eq. 21) in response to periodic mean modulations of
of different amplitudes. For an illustration of how the population dynamics
emerges in response to a dynamic stimulus see Fig. 3B.
Figure 7.
Weak spike correlations in the threshold model.
(Top) Illustration of spike correlations resulting from common input that are studied in A and B. (A) Cross conditional firing rate vs. time
in the limit of weak input correlations
. Both neurons have the same voltage correlation function
, firing rate
. Fixed firing rate and varying correlation times
(A, left) or fixed correlation time
and varying firing rates
(A, right). (B) Peak spike correlation
as a function of firing rate
. Symbols denote the corresponding peak spike correlations from (A).
Figure 8.
Statistics of spike triggering events in the threshold neurons.
(A) Spike triggered average voltage for
and firing rates
; simulated results (circles) and analytical solution in Eq. 37 (solid lines). (B) Spike triggered voltage covariance
for
,
, the cross section
is shown at the right. Simulated results (circles) and analytical solution in Eq. 39 (solid lines). The solid vertical black line indicates
.
Figure 9.
Demonstration of population firing rate modulation and phase locking.
(A) Simulated population firing rate for mean current modulation for
,
,
and
, time bin
. This results in
,
and in the amplitude of the firing rate modulation of
. Solid lines denote the envelop of
(red) and the current modulation (black). Black and red arrows indicate the phase relation between the input current and the evoked firing rate response. (B) Theoretical distribution of phase lags
for varying modulation depth
, for illustration we chose
(from (A)). The solid curves are the distribution envelop for
(red),
(black),
(blue). The arrows indicate the corresponding mean phase
.