Inferring Neural Connectivity from Multiple Spike Trains
Seung Kee Han, Dong-Uk Hwang and Seon Young Ryu
Department of Physics, Chungbuk National University; skhan@chungbuk.ac.kr
Abstract: Recently the temporal coding based on the
spike timing is one of the hot issues of the
neuroscience. In the neural network, the spike timing
depends on the external stimulus and also on the
internal network structure. In this paper, we propose a
method of inferring network connectivity from the
multiple spike trains. It is based on the phase model
description of the spike trains. A continuous phase
variable is introduced for each of the spike trains by
assigning 2 phase for each of the spike intervals and
by the linear interpolation. The relative strength of
the mutual dependence allows us to estimate the
relative strength of the coupling as well as the type of
the coupling. We report the result of our test on the
coupled neural network model and also on the
electronic circuit experiment. When compared with
the conventional method based on the crosscorrelogram, the proposed method is much more
effective in estimating the network connectivity. At
the same time, the measurement of the effective
coupling allows to estimate the type of coupling
Introduction: Using the multi-channeling recording,
it becomes possible to record simultaneously the
spike trains of hundreds or thousands of neurons.
The conventional correlation-based analysis, like the
cross-correlogram in time domain or the mutual
coherence in frequency domain have the most
widespread use in identifying functional relationship
between two group of neurons. However, it has
difficulties in distinguishing correlated activities due
to direct connectivity and that of indirect connectivity.
On the other, it was shown recently that the phase
model description of nonlinear oscillators is very
effective in unveiling mutual coupling underlying
chaotic physiological data.[1] The phase model
description is also useful for the analysis of coupled
stochastic oscillators.[2] Motivated by these
developments, we propose a dynamic phase model
analysis for a discrete spike trains. When the
variability of the spike timing is not so strong, the
multiple spike trains are converted into continuous
phase variables. Reduction of the complexity of the
dynamic variables will make the analysis of the
mutual dynamic dependences between phase
variables much simpler. As a consequence, it will be
possible to infer the type of neural connectivity and
also the network connectivity. This method will be
applied to spike trains obtained from the neural
network model and the electronic circuit experiment.
Phase Analysis: A continuous phase variable is
generated for each of the spike trains of neurons. As
shown in Fig.1, a phase shift of 2 is assigned for
each of the spike train, irrespective of the interval
length. In between the spike event, the phase is
interpolated. In terms of phase variables 1 and 2
for the neuron N1, and N2 respectively, a simple
model replaces the interaction between two
oscillatory neuron as:
d(1)/dt= F1(1,2)= f1(1) + G1(1, 2) + 1
d(2)/dt= F2(2,1)= f2(2) + G2(2, 1) + 2
(1)
Here F1(1,2) and F2(2,1) denote the phase
velocity for the neuron N1 and N2 respectively. The
phase velocity F1(1,2) of the neuron N1 is
decomposed into three contributions: f1(1), the
phase velocity for the uncoupled neuron N1,
G1(1,2), the contribution due to synaptic coupling
of the neuron N2, and finally the the noise terms 1.
Similarly, the phase velocity F2(2,1) for the neuron
N2 is decomposed into three terms.
1
2
2
2
2
2
2
Fig.1 The spike trains for neuron N1 and N2. For
each spike train, a phase shift of 2 is assigned
irrespective of the inter-spike interval.
2
1(1,2)
2

2(2)
1
2
0
0
Fig.2 The phase portrait of the 1 and 2. The
evolution for a time interval gives the phase shifts
1(1,2) and 2(2,1), when divided by the time
interval  gives the phase velocity F1(1,2) and
F2(2,1). This figure illustrates the dependence of
F1(1,2) on 2, while no dependence of F2(2,1)
on 1.
For estimation of the phase velocity F1 and F2,
evolve the phase 1 and 2 for a time interval  as
shown in Fig.2 and then divide the phase shifts by .
Approximation by a few harmonics of lower order
gives the phase velocity F1(1,2) and F2(2,1) as
shown in Fig3. As the dependence of F1(1,2) on
the phase 2, we take the partial derivative F1/2,
and similarly for the dependence of F2(2,1) on
the phase 1, take the F2/1. For a simple phase
model in Eq.1, the dependence of phase velocity
F1(1,2) on phase 2 gives an estimation of the
coupling term G1(1,2). Likewise the dependence
of F2(2,1) on the phase 1 represents the
estimation of the coupling term G2(2,1). Therefore
the comparison of the amplitude of the F1/2 and
F2/1 will allow us to estimate the strength of
G1(1,2) and G2(2,1), the strength of the
coupling W21 and W12. Refer to the ref.[3].
Fig. 3 The plot of phase velocity F1(1,2) and the
plot of F1/2, the dependence of the phase
velocity F1(1,2) on the phase 2. Similar plots for
the neuron N2 give the dependence of phase velocity
F2(2,1) on the phase 1.
Results: As a test of the proposed method, we
analyze the spike trains obtained from an electronic
circuit that emulates the coupled coherent resonance
oscillators, shown in Fig.4. Each of the coherent
resonance oscillator units is driven by the
independent noise to generate quasi-periodic spiking
pattern.[2]
The coupling between two units is mediated by the
two uni-directional resistances. In fig.4, the coupling
strength g12 is fixed at 0.01 with g21 changing.
For various values of the coupling strength g21,
the amplitude of F1/2 is denoted as a circle. It
grows with the coupling strength g21, while the
amplitude of the F1/2, denoted as a dagger,
remains nearly constant like the coupling strength
g12. This result implies that the measurement of
the phase dependence between two phase variables is
a good measure for the estimation of the relative
strength of the coupling.
Fig.5 The plot of phase dependence F1/2
(denoted by a dot) and the F2/1 (denoted by a
dagger).
When the coupling term G1(1,2) is re-expressed as
G1(1,2-1), the dependence on the 1 is weak
when the coupling is weak. Neglecting the weak
dependence on the 1, the dependence of coupling
term G1(1,2-1) on the phase difference 2-1
will be denoted as G1(-), an effective coupling.
Likewise, the effective coupling for N2 is represented
as G2(). The plot of the effective coupling as a
function of the phase difference is very informative in
the analysis of the phase dynamics between two
coupled oscillators.[4] For two identical oscillators
with symmetric coupling, the evolution of the phase
difference  is governed by the G()=G2()-G1(), as d/dt = G().
V
o
2
Fig.4 An electronic circuit that emulates a coupled
coherent resonance oscillators.
In Fig.5, we plot the effective coupling -G() for
the synaptic coupling for various values of the
synaptic reversal potential V_syn. The slope at the
fixed point determines the stability of the phase
locked solution. For the excitatory coupling, the
stability of the in-phase solution revealed as positive
slope at =0. While for inhibitory coupling, a stable
anti-phase solution is observed
We test this for a network consisted of 12 neurons
(with 9 excitatory neurons and 3 inhibitory neurons)
fully connected with each other. Using spike trains of
12 neurons, we calculate the effective coupling for
each pair of the neurons. From the strength of the
effective coupling, we estimate the network
connectivity. In Fig.7, the estimated coupling strength
is plotted versus the pre-assigned coupling strength in
the neural network. A strong positive correlation
between these two quantities is observed. While in
Fig.7(b), no correlation between the crosscorrelogram and the coupling strength is observed.
Fig.5 The plot of effective coupling for various
values of the synaptic reversal potential V_syn. When
V_syn is bigger than the resting potential V_rest, the
coupling is of excitatory type and the in-phase
solution is stable with the slope at =0 positive.
While for inhibitory coupling with V_syn smaller than
V_rest, the anti-phase solution is stable.
For a non-symmetric coupling, the plot of the odd part
of the effective coupling G1(-) and G2() also
allows us to understand the role of the coupling on
the phase dynamics. In Fig.6, we plot the odd part of
the effective coupling G1(-) for various values of
the coupling strength g21 with g12 fixed. From
the shape of the effective coupling G1(-), the role
of excitatory coupling even for an asymmetric
coupling is very evident. At the same time, the
increase of the effective coupling with the coupling
strength g21 indicates that the strength of the
effective coupling G1(-) together with the phase
sensitivity F1/2 becomes a measure good for the
estimation of the coupling strength.
Fig.7 The plot of the strength of the effective
coupling (a) and the cross-correlogram (b) versus the
coupling strength.
Conclusions: A phase model description of the
multiple spike trains is proposed. It is shown that the
calculation of the mutual phase dependence is a good
measure that allows us to infer the network
connectivity and the effective coupling is very
informative in understanding the phase dynamics.
We have shown that this analysis works very well for
spike trains obtained from an electronic circuit
experiment and neural network model. We expect that
this method will be also very useful for the analysis
of multiple spike trains.
[1] M.G. Rosenblum and A. Pikovsky, J. Kurths, Phys.
Rev. Lett. 76, 1804 (1996).
[2] S.K. Han, et. al., Phys. Rev. Lett. 83, 1771 (1999).
[3] M.G. Rosenblum and A. Pikovsky, Phys. Rev. E 64,
0455202 (2001).
[4] S.K. Han, C. Kurrer, and Y. Kuramoto, Phys. Rev.
Lett. 75, 3190 (1995).
Fig.6 The plot of the effective coupling for various
values of the coupling strength.