1. 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, (2004)
Riecke, Roxin, Madruga, Solla - Chaos 17, (2007)

2. Complex Networks: Form to Function
To which extent does network topology determine or affect network function?

3. 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)

4. 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, (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, (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, (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)].

5. Dentate Gyrus: a Small-World Network
Complex network topology: neither regular, nor random
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)]

6. Excitable Integrate-and-Fire Neurons
Spikes are produced whenever:
Followed by a reset:
Excitable neurons if:
<

7. Network Activity: p = 0

8. Network Activity: p = 0.05

9. Sustained Network Activity
p=0
p=0.05

10. Instantaneous Firing Rate, p = 0.10

11. Sustained Network Activity: Oscillations

12. Network Activity: p = 0.15

13. Sustained Network Activity
p=0.05
p=0.15

14. Failure to Sustain Oscillations
100
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?

15. 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)]

16. 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.

17. Transition to Failure

18. Sustained Oscillations: Backbone Pathway
Backbone neurons shown in red; N=1000, p=0.10, and TR =

19. 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
Number of attractors vs N

20. Fast Waves: D= 0.10
p = 0.01, 0.05, 0.10, 0.15, 0.20, 0.25, from (a) to (f)

21. 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

22. Slow Waves: Increasing D 
D= 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18

23. Reentrant Network Activity: p = 0.8
D= 0.16

24. Reentrant Network Activity: p = 1.0
D= 0.16

25. 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.

26. Chaotic Neural Activity
Temporal complexity of activity patterns in chaotic regime.
N=1000, D= 0.18
Lifetime of chaotic activity: stretched exponentials.
p=1, D= 0.18 and D= 0.165

27. Bistability: a switch-off mechanism

28. Summary CONNECTIVITY MATTERS!!!
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!!!