Figure 1.
A pair of integrate-and-fire model neurons driven by partially shared and correlated presynaptic events.
A Each of the neurons and
receives input from
sources, of which
are excitatory and
are inhibitory. Both neurons share a fraction
of their excitatory and inhibitory sources, whereas the fraction
is independent for each neuron. Schematically represented spike trains on the left of the diagram show the excitatory part of the input; the inhibitory input is only indicated. A single source emits spike events with a firing rate
, with marginal Poisson statistics. Correlated spiking is introduced in the
common excitatory sources to both neurons. This pairwise correlation is realized by means of a multiple interaction process (MIP) [39] that yields a correlation coefficient of
between any pairs of sources. In absence of a threshold, the summed input drives the membrane potential to a particular working point described by its mean
and standard deviation
and the correlation coefficient
between the free membrane potentials
,
of both neurons. In presence of a threshold mean and variance of the membrane potential determine the output firing rate
and their correlation in addition determines the output correlation
, calculated by (2). B–E Direct simulation was performed using different values of common input fraction
and four fixed values of input spike synchrony
(as denoted in E). Each combination of
and
was simulated for
seconds; gray coded data points show the average over
independent realizations. Remaining parameters are given in Table 1. Solid lines in B and C are calculated as (5) and (6), respectively. In C, for convenience,
is normalized by the common input fraction
, so that
in absence of synchrony (
). E shows the output spike synchrony
.
Figure 2.
Isolation and control of the effect of synchrony on the free membrane potential statistics.
A,B Adjusted common input fraction (A) and input firing rate
(B) for different values of
(gray coded) that ensure the same variance and covariance as for
. C Correlation coefficient
normalized by
between the free membrane potential of a pair of neurons using the adjusted common input fraction
. D Standard deviation of the free membrane potential, using the adjusted firing rate
. The statistics of the free membrane potential measured in simulations in panels C and D are further verified via (6) and (5) (solid lines).
Figure 3.
Correlation transmission of a pair of integrate-and-fire neurons.
A Output spike synchrony as a function of input correlation and for four different values of input synchrony
,
,
and
(gray-coded). Dashed black line with slope
indicates
. B Corresponding mean output firing rate of the neurons. C,D Cross-correlation functions at input correlations
(C) and
(D) (indicated by dashed vertical lines in A) for the three values of input synchrony
as indicated in C.
Figure 4.
Approximation of the output correlation in the limit of low input correlation.
A Correlation transmission in the low input correlation limit. Data points show the output correlation resulting from simulations, solid lines show the theoretical approximation
(10). Dashed black line indicates
. B Deflection of the firing rate with respect to base rate caused by an additional synaptic event at
.
Figure 5.
Neural dynamics in the regime of high input correlation and strong synchrony.
A Exemplary time course of a membrane potential driven by input containing strong, synchronous spike events. During the time period shown, five MIP events arrive (indicated by tick marks above ). The first four drive the membrane potential above the threshold
, after which
is reset to
and the neuron emits a spike (dark gray tick marks above
). The fifth event is not able to deflect
above threshold (light gray) and the membrane potential quickly repolarizes towards its steady state mean
(see text). B Time-resolved membrane potential probability density
triggered on the occurrence of a MIP event at
. Since most MIP events elicit a spike, after resetting
to
the membrane potential quickly depolarizes and settles to a steady state Gaussian distribution. The slight shade of gray observable for small
just below the threshold
is caused by the small amount of MIP events that were not able to drive the membrane potential above threshold. C Probability density of the membrane potential in steady state. Theoretical approximation (black) was computed using
and
(see text and eq:Vt), empirical measurement (gray) was performed for
(gray dashed line in B). Simulation parameters were
,
(
) and
(
) Hz. Other parameters as in Table 1.
Figure 6.
Approximation of the output synchrony in the limit of high input correlation.
A Output spike synchrony as a function of input correlation in the limit of high input correlation and strong synchrony . Data points and solid lines show results from simulations and theoretical approximation (11), respectively. Gray code corresponds to the four different mean membrane potential values
as depicted in B, the input firing rate
was
,
,
and
, correspondingly (from low to high
). The working point used in the previous sections corresponds to
,
. The inset shows the output firing rate at the four working points. B Output spike synchrony as a function of the actual common input fraction
at the four working points. Dashed curves in A and B indicate
.
Figure 7.
Correlation transmission for the output correlation on a long time scale in the presence of strong input synchrony (
).
A,B As Fig. 6A,B but measuring the spike count correlation between the neurons over a time window of . C Mean
(thick lines) and mean plus minus one standard deviation
(thin lines) of the amplitude of synchronous spike volleys in the common excitatory input as a function of
for three different values of
(indicated by gray code). D Mean
and standard deviation
of the membrane potential caused solely by the disjoint afferents for strong synchrony (
) as a function of
.
Figure 8.
Correlation transmission for strong synchrony () with jittered spike volleys.
Panels show simulation results using and four different jitter widths
,
,
and
(gray code as shown in panel A). A Output firing rate as a function of input correlation for different jitter widths. B–D Output correlations
(B),
(C) and
(D) as a function of the input correlation for increasing jitter widths.
Figure 9.
Mechanistic model of enhanced correlation transmission by synchronous input events.
A The detailed model discussed in the results section is simplified two-fold. 1) We consider binary neurons with a static non-linearity . 2) We distinguish two representative scenarios with different models for the common input:
: Gaussian white noise with variance
, representing the case without synchrony, or
: a binary stochastic process
with constant amplitude
, mimicking the synchronous arrival of synaptic events. In both scenarios in addition each neuron receives independent Gaussian input. B Marginal distribution of the total input
to a single neuron for input
(gray) and
(black) and for
. In input
the binary process
alternates between
(with probability
) and
(with probability
), resulting in a bimodal marginal distribution. The mean activity of one single neuron is given by the probability mass above threshold
. We choose the variances
and
of the disjoint Gaussian fluctuating input such that the mean activity is the same in both scenarios. C Output correlation
as a function of the input correlation
(see A) between the total inputs
and
. Probability
is chosen such that inputs
and
result in the same input correlation
. The four points marked by circles correspond to the panels D–G. D–G Joint probability density of the inputs
,
to both neurons. For two different values of
the lower row (E,G) shows the scenario
, the upper row (D,F) the scenario
. Note that panel B is the projection of the joint densities in F and G to one axis. Brighter gray levels indicate higher probability density; same gray scale for all four panels.
Table 1.
Parameters of the input and LIF neuron used in the simulations.