Fast and Slow Dynamics in Neural Networks with Small-World Connectivity

Fast and Slow Dynamics in Neural Networks with Small-

World Connectivity

Roxin, Riecke, Solla - Phys. Rev.Lett. 92, 198101 (2004)Riecke, Roxin, Madruga, Solla - Chaos 17, 026110 (2007)

With: Santiago Madruga, Hermann Riecke, Alex Roxin

Sara A. SollaNorthwestern

University

Complex Networks: Form to Function

To which extent does network topology determine or affect network function?

Model of Network Connectivity: a Small-

World Network Many complex networks have a small-world topology characterized by dense local clustering or cliquishness of connections between neighboring nodes yet a short path length between any (distant) pair of nodes due to the existence of relatively few long-range connections.

This is an attractive model for the organization of brain anatomical and functional networks because a small-world topology can support both segregated (specialized) and distributed (integrated) information processing.

Bassett, Bullmore - The Neuroscientist 12, 512 (2006)

Small-World (SW) Brain Networks

• Activity in hippocampal slices has been successfully modeled using SW networks of excitatory neurons that reproduce both bursts (CA3) and seizures (CA1) [Netoff, Clewley, Arno, Keck, White - J. Neurosci. 24, 8075 (2004)].

• Large-scale synchronization associated with epileptic seizures has been modeled using SW networks of Hindmarsh-Rose neurons [Percha, Szakpasu, Zochowski, Parent - Phys. Rev. E 72, 031909 (2005)].

• Small-world networks are increasingly being applied to the analysis of human functional networks derived from EEG, MEG, and fMRI experiments [Eguiluz, Chialvo, Cecchi, Baliki, Apkarian - Phys. Rev. Lett. 94, 018012 (2005)].

• The aggregate system of neurons and glial cells can be viewed as a small-world network of excitable cells [Sinha, Saramaki, Kaski - Rev. E 76, 015101 (2007)].

• A large-scale structural SW model of the dentate gyrus has been formulated and used to identify topological determinants of epileptogenesis [Dyhrfjeld-Johnsen, Santhakumar, Morgan, Huerta, Tsimring, Soltesz - J. Neurophysiol. 97, 1566 (2007)].

Dentate Gyrus: a Small-World Network

Complex network topology: neither regular, nor random

QuickTime™ and aTIFF (Uncompressed) decompressor

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L : average path length C : clustering coefficient

The average number of synapses between any two neurons in the dentate gyrus is less than three - similar to the average path length for the nervous system of C. Elegans, which has only 302 neurons as opposed to over one million! [Dyhrfjeld-Johnsen, Santhakumar, Morgan, Huerta, Tsimring, Soltesz - J. Neurophysiol. 97, 1566 (2007)]

Excitable Integrate-and-Fire Neurons

Spikes are produced whenever:

Followed by a reset:

Excitable neurons if:

<

Network Activity: p = 0

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Network Activity: p = 0.05



Sustained Network Activity

p=0 p=0.05

Instantaneous Firing Rate, p = 0.10

Sustained Network Activity: Oscillations



Network Activity: p = 0.15



Sustained Network Activity

p=0.05 p=0.15

Failure to Sustain Oscillations

100

0

Ensemble average over many network configurations with the same density p of shortcuts. Some of the configurations will sustain persistent oscillatory activity, while some will burst and fail. Is there a well defined transition for large networks?

Failure to Sustain Oscillations

Failure involves the interaction of two time scales:

1) A cellular time scale associated with the time TR needed for a neuron to recover to the point where a single synaptic input will make it fire:

2) A network time scale associated with the TN (p) for the first return of activity in a small-world network:

[Newman, Moore, Watts - Phys. Rev. Lett. 84, 3201 (2000)]

, where

Transition to Failure

The failure transition occurs at a size-dependent critical density of shortcuts, determined by the condition

The critical density pcr scales with the logarithm of the size N of the system.

Transition to Failure

Sustained Oscillations: Backbone Pathway



Backbone neurons shown in red; N=1000, p=0.10, and TR = 2.494.

Sustained Oscillations: Attractors

Oscillatory solutions characterized by their period, their mean firing rate, and the standard deviation of their firing rate.

N=1000, p=0.05, D= 0.10


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Number of attractors vs N

Fast Waves: D= 0.10



p = 0.01, 0.05, 0.10, 0.15, 0.20, 0.25, from (a) to (f)

Slow Waves: D= 0.16



p = 0.01, 0.20, 0.40, 1.00, from (a) to (d)

Activity in (d) is noisy and exhibits synchronized population spikes

Slow Waves: Increasing D

D= 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18

Reentrant Network Activity: p = 0.8

D= 0.16

Reentrant Network Activity: p = 1.0

D= 0.16

Chaotic Neural Activity

N=1000, D= 0.16



Dashed vertical line indicates TR, the minimum value of the interspike interval (ISI) if neurons receive only one input per cycle. As p increases, an increasing number of neurons exhibit ISIs below TR. These ‘faster’ neurons receive multiple inputs via shortcuts, and they sustain network activity while the `slower’ neurons recover.

Chaotic Neural Activity

Temporal complexity of activity patterns in chaotic regime.

N=1000, D= 0.18



Lifetime of chaotic activity: stretched exponentials.QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.



p=1, D= 0.18 and D= 0.165

Bistability: a switch-off mechanism

Summary• Small-world networks of excitable neurons are capable of supporting sustained activity. This activity is sparse and oscillatory, and it does not require excitatory-inhibitory interactions.

• A transition to failure occurs with increasing density of shortcuts. Below the failure transition, the number of attractors increases at least linearly with the size N of the system. A connectivity backbone can be associated with each attractor.

• Above the transition, the network dynamics exhibit exceedingly long chaotic transients; failure times follow a stretched exponential distribution. Periods of low activity are mediated by `early firing’ neurons that receive more than one shortcut input. This chaotic activity does not require a balanced excitatory-inhibitory network.

CONNECTIVITY MATTERS!!!

Fast and Slow Dynamics in Neural Networks with Small-World Connectivity

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