Percolation in Living Neural Networks Ilan Breskin, Jordi Soriano , Tsvi Tlusty, and Elisha Moses
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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. 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. 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 p s to belong to a cluster of size s-1. Experimental results – Giant component Experimental setup – Network response 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. Experimental results – Cluster distribution analysis Numerical simulations of the model Study of the balance between excitation and inhibition Summary and conclusions 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 network response is studied in terms of percolation on a graph. We show that the network undergoes a percolation transition at a critical connectivity c 0 , 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. 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) I. Breskin, J. Soriano, T. Tlusty, and E. Moses, Percolation in Living Neural Networks, Phys. Rev. Lett. 97, 188102 (2006). (2) J.-P. Eckmann, O. Feinermann, L. Gruendlinger, E. Moses, J. Soriano, and T. Tlusty, The Physics of Living Neural Cultures, Phys. Reports, in press. In percolation theory, the critical point c 0 depends on the average number of connections per neuron. For the excitatory network, c e ~ 1/n e , while for the excitatory-inhibitory network, c ei ~ 1/(n e - n i ).The ratio between excitation and inhibition is then given by For hippocampal (HPC) cultures: n i = 0.33 n e . For cortical (CTX): n i = 0.17 n e . (a) Fluorescence image of a small region of the neural culture. The neural culture contains ~10 5 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. vel approach: collective electric stimulation + gradual weakening of the network. Without giant component: 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: By fitting a polynomial p s x s to H(x) we can extract information of the input- clusters distribution in the neural culture. 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. (a) Cluster size distribution p s (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 p s to loops. Left: neurons forming a chain--like connectivity give a p s distributed uniformly. Center: closing the loop by adding just one link collapses p s to a single peak. Right: additional links increase the average connectivity <k>, but do not modify p s . 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 biggest jump in the response curve (V) corresponds to a “firing” area that extends across the entire culture. Low clusterization High clusterization 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 p s (s) are obtained when loops are introduced in the network. ) a (
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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. Abstract - PowerPoint PPT Presentation
Transcript of Percolation in Living Neural Networks Ilan Breskin, Jordi Soriano , Tsvi Tlusty, and Elisha Moses
Percolation in Living Neural NetworksIlan Breskin, Jordi Soriano, Tsvi Tlusty, and Elisha Moses
Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, Israel.
AbstractWe 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.
Percolation modelWe 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.
Experimental results – Giant component
Experimental setup – Network response
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.
Experimental results – Cluster distribution analysis
Numerical simulations of the model
Study of the balance between excitation and inhibition
Summary and conclusionsWe 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 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.
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) I. Breskin, J. Soriano, T. Tlusty, and E. Moses, Percolation in Living Neural Networks, Phys. Rev. Lett. 97, 188102 (2006). (2) J.-P. Eckmann, O. Feinermann, L. Gruendlinger, E. Moses, J. Soriano, and T. Tlusty, The Physics of Living Neural Cultures, Phys. Reports, in press.
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
For hippocampal (HPC) cultures: ni = 0.33 ne. For cortical (CTX): ni = 0.17 ne.
(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.
Novel approach: collective electric stimulation + gradual weakening of the network.
Without giant component:
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:
By fitting a polynomial psxs to H(x) we can extract information of the input-clusters distribution in the neural culture.
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.
(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.
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 biggest jump in the response curve (V) corresponds to a “firing” area that extends across the entire culture.
Low clusterization
High clusterization
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.
(a)
