Fast and Slow Dynamics in Neural Networks with Small-World Connectivity
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Transcript of 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
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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:
<
Sustained Network Activity: Oscillations
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Failure to Sustain Oscillations
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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.
Sustained Oscillations: Backbone Pathway
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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
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p = 0.01, 0.05, 0.10, 0.15, 0.20, 0.25, from (a) to (f)
Slow Waves: D= 0.16
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p = 0.01, 0.20, 0.40, 1.00, from (a) to (d)
Activity in (d) is noisy and exhibits synchronized population spikes
Chaotic Neural Activity
N=1000, D= 0.16
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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
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Lifetime of chaotic activity: stretched exponentials.QuickTime™ and a
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p=1, D= 0.18 and D= 0.165
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!!!