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Fast and Slow Dynamics in Neural
Networks with Small-World Connectivity
Sara A. Solla
Northwestern University
With: Santiago Madruga, Hermann Riecke, Alex Roxin
Roxin, Riecke, Solla - Phys. Rev.Lett. 92, 198101 (2004)
Riecke, Roxin, Madruga, Solla - Chaos 17, 026110 (2007)
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 smallworld 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 smallworld 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
L : average path length
C : clustering coefficient
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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
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Sustained Network Activity
p=0
p=0.05
Instantaneous Firing Rate, p = 0.10
Sustained Network Activity: Oscillations
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Network Activity: p = 0.15
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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:
, where
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)]
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
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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.
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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
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
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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
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Temporal complexity of activity
patterns in chaotic regime.
N=1000, D= 0.18
Lifetime of chaotic activity:
stretched exponentials.
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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!!!