Small World Networks


Abstract:

In hippocampal slice models of epilepsy, two behaviors are seen: short bursts of electrical activity lasting 100 ms, and seizure-like electrical activity lasting seconds.  The bursts originate from the CA3 region, where there is a high degree of recurrent excitatory connections.  Seizures originate from the CA1, where there are fewer recurrent connections.  In attempting to explain this behavior we simulated model networks of excitatory neurons, using several types of model neurons.  The model neurons were connected in a ring containing predominantly local connections and some long distance random connections, resulting in a small-world network connectivity pattern.  By changing parameters such as the synaptic strengths, number of synapses per neuron, proportion of local versus long distance connections, we induced “normal”, “seizing,” and “bursting” behaviors.  Based on these simulations, we made a simple mathematical description of these networks, under well-defined assumptions.  This mathematical description explains how specific changes in the topology or synaptic strength in the model cause transitions from “normal” to “seizing” and then to “bursting”.  These behaviors appear to be general properties of excitatory networks.

 

A full pdf of this paper to appear in the Journal of Neuroscience can be downloaded here.

Introduction:

Small world networks are sets of nodes connected with mostly local connections and a few randomly connected long distance connections.  These long distance connections reduce the number of synapses traversed to connect any pair of neurons in the network.  We use small world networks to model the brain because it has some structure, but a free parameter to change the orderliness in the network.  Below are examples of networks with progressively more randomness in the connections as you progress to the right.

 

Using various models of neurons, noisy integrate-and-fire neurons, stochastic Hodgkin-Huxley neurons, and Poisson firing models with probabilistic response to synatpic inputs, connected in a small-world network fashion, we have observed different forms of epileptiform behaviors by changing properties of the network.

Below is a set of figures illustrating these behaviors.

Movie of how network is mapped into a ring (1.55 MB AVI).

Movie of Seizure (2.1 MB AVI))

3000 Neurons
30 Synapses per neuron
0.5% of synapses long distance


Movie of Bursting (1.4 MB AVI)

3000 Neurons
30 Synapses per neuron
20% of synapses long distance


Movie of Seizure (2.3 MB AVI)

3000 Neurons
90 Synapses per neuron
0.1% of synapses long distance


Movie of Bursting (2 MB AVI)

3000 Neurons
90 Synapses per neuron
1% of synapses long distance

Still frames of these movies are shown below.  Panel C shows still frames from 3000 neurons with 30 synapses and 0.5% of the synapses randomly connected.  This behavior shows sustained high activity we liken to seizures  Panel D shows stills of a network with 3000 neurons with 90 synapses each and 20 percent of the synapses set at 20%.  This population activity oscillates and we liken to bursting behavior.

Below it can be seen how the network behavior changes as the number of long distance synapses in the network is varied.  Where the transition from normal to seizing and seizing to bursting occurs is different for networks with different number of synapses per neuron.

Mathematical equations can be used to describe these behaviors.  Using these equations, we can predict the behaviors of excitatory networks based on the dynamics of the neurons.  Belo is a graphical representation of the new wave rate and the wave death rate as a function of the number of waves present in the network.  Where the two rates are equal is an equilibrium   This equilibrium is stable and attracting as long as the new wave rate is not too high (left and middle panels).  When the slope of the new wave rate is greater than the death rate at the equilibrium, the network becomes unstable and bursting behavior is seen (right panels).

Below is shown how the network behaviors change as the number of  synapses per neuron are varied and the number of random connections is changed (right panel) or the strength of the synapse (left).  The gray lines indicate values for which network simulations were run.  The letters indicate the values for which examples are shown above.  Vertical bars show where the transitions were observed in the simulations.

 

A full pdf of this paper to appear in the Journal of Neuroscience can be downloaded here.

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Last updated: July 13, 2004

Theoden Netoff
Center for Biodynamics
Postdoctoral Fellow
Boston University