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Percolation in Living Neural Networks
Ilan Breskin, Jordi Soriano, Tsvi Tlusty, and Elisha Moses
Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, Israel.
Experimental results – Giant component
Abstract
We study neural connectivity in cultures of rat hippocampal neurons, and extract statistical
properties of the underlying network. The neurons are excited by a global electrical
stimulation applied to the entire network through bath electrodes. Gradual addition of CNQX
blocks the neuro-receptors AMPA and decreases the neural connectivity. The process of
disintegration of the network is described in terms of percolation on a graph, yielding a
quantification of the connectivity in the network. With no CNQX the network comprises of
one big cluster (giant component). Increasing the CNQX concentration, the network
fragments into smaller clusters and the connectivity undergoes a percolation transition,
described by a critical exponent β = 0.65 ± 0.05. The exponent β is independent of the balance
between excitatory and inhibitory neurons. Together with numerical simulations we show that
the neural connectivity corresponds to a local, Gaussian connectivity rather than a power law
one.
Numerical simulations of the model
Numerical simulations of the model give response curves that reproduce the experimental ones. An exponent  = 0.65
is obtained for a Gaussian degree distribution of neural connections. A power law distribution gives  = 1. The
simulations also show that the peak distributions in ps(s) are obtained when loops are introduced in the network.
Study of the giant component as percolation transition. The main plot shows the size of the giant component as
a function of the synaptic strength (connectivity) c, for a network containing both excitatory and inhibitory
neurons (black dots) and a network containing excitatory neurons only (blue squares). Inhibition is blocked with
bicuculine. Some CNQX concentrations are indicated for clarity. Inset: Log—log plot of the power law fits g ~
|1-c/co|. The slope 0.65 corresponds to the average value of  for the two networks.
High clusterization
Experimental setup – Network response
Novel approach: collective electric stimulation + gradual weakening of the network.
(a)
Low clusterization
Study of the balance between excitation and inhibition
Response curves (V) for 6 concentrations of
CNQX. The grey bars show the size of the giant
component.. Inset: Corresponding H(x) functions
(see model). The bar shows the size of the giant
component for 300 nM.
(a) Fluorescence image of a small region of the neural culture. The neural culture contains ~105 neurons, and
we monitor the activity of a small region containing ~600 neurons. (b) Collective stimulation through bath
electrodes. (c) Activity plot of the neural response. Black lines indicate those neurons that respond to the
excitation. (d) Fluorescence signal for 3 neurons at increasing voltages. Vertical lines show the excitation
time, and arrows the responding neurons.
In percolation theory, the critical point c0 depends on the average
number of connections per neuron. For the excitatory network, ce ~ 1/ne ,
while for the excitatory-inhibitory network, cei ~ 1/(ne - ni ).The ratio
between excitation and inhibition is then given by
Experimental results – Cluster distribution analysis
(a) Cluster size distribution ps(s) for the generating functions H(x) shown in the left panel. The values in the table
indicate the concentration of CNQX for each curve. The distribution is characterized by a peaked distribution
that persists even for relatively high concentrations of CNQX, suggesting that loops and strong locality may be
present in the neural culture. (b) Sensitivity of peaks in ps to loops. Left: neurons forming a chain--like
connectivity give a ps distributed uniformly. Center: closing the loop by adding just one link collapses ps to a
single peak. Right: additional links increase the average connectivity <k>, but do not modify ps.
For hippocampal (HPC) cultures: ni = 0.33 ne.
For cortical (CTX): ni = 0.17 ne.
Percolation model
We consider a simplified model of the network in terms of bond-percolation on a graph. A neuron has a
probability f = f(V) to fire as a direct response to the externally applied electrical stimulus, and it always fires if
any one of its input neurons fire. A neuron has a probability ps to belong to a cluster of size s-1.
Summary and conclusions
Without giant component:
By fitting a polynomial psxs
to H(x) we can extract
information of the inputclusters distribution in the
neural culture.
With a giant component present, the population is divided between the neurons the belong to the giant
component and the ones that belong to finite clusters:
We present a novel experimental technique to extract statistical properties of living neural
networks. We study the network’s response to an electric stimulation applied to the entire
culture, and for gradual reduction of the connection strength between neurons.
The analysis of the spatial distribution of
the neural response indicates that the
neurons in denser areas (higher number of
connections) tend to fire first in response to
the external excitation. The neurons that
respond next are the immediate neighbors.
The network response is studied in terms of percolation on a graph. We show that the
network undergoes a percolation transition at a critical connectivity c0, characterized by a
critical exponent   0.65 that is independent of the balance between excitation and
inhibition. Numerical simulations show that this value corresponds to a Gaussian degree
distribution and not a power law one.
The biggest jump in the response curve
(V) corresponds to a “firing” area that
extends across the entire culture.
In conclusion, the results indicate that the connectivity in neural cultures is local,
characterized by a Gaussian degree distribution with a considerable presence of clusters.
References
(1)
(2)
I. Breskin, J. Soriano, T. Tlusty, and E. Moses, Percolation in Living Neural Networks, Phys. Rev. Lett. 97, 188102 (2006).
J.-P. Eckmann, O. Feinermann, L. Gruendlinger, E. Moses, J. Soriano, and T. Tlusty, The Physics of Living Neural Cultures, Phys. Reports, in press.