Spatiotemporal Networks in Addressable Excitable Media

International Workshop on

Bio-Inspired Complex Networks in Science and TechnologyMax Planck Institute for the Physics of Complex Systems

Dresden, Germany, 5 - 9 May 2008

Co-Workers:Mark TinsleyAaron Steel

Funding:NSFONRW.M. Keck Foundation

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Ru(bipy)3

2+ + BrMA + hν → Br -

Photosensitive BZ Network

Dark system is oscillatory.

Excitability of medium maintained by reference light intensity.

Medium is divided into an array of cells.

Non-diffusive jumps from one cell to another (random) cell.

Nearest-neighbor interactions described by reaction-diffusion wave.

Long range interactions described by non-diffusive jumps.Phys. Rev. Lett. 95, 038306 (2005)Chaos 16, 015110 (2006)

Static Link Network

Local links occur via reaction-diffusion wave.

Non-local links occur via reduction of light intensity at destination cell once a source cell reaches threshold.

Threshold: 50% excited.

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Network Architecture

Mean path length: the average minimum separation of two nodes within the network.

Clustering coefficient: the average number of links between neighbors of an individual node divided by the total possible number of links between these neighbors.

Small and large number of links: values reflect the lattice and random graph.

Intermediate number links: values typical of small-world networks.

Static Network with 10 Links

Oregonator Model Simulations

Successive snapshotsfrom left to right

Grid: 1000 × 1000 Cells: 20 × 20 grid pts50 × 50 = 2500 cells

Threshold: 50% excited

Static Network with 100 Links

Static Network with 500 Links

Static Networks: Links

10 links 100 Links 500 Links

Experimental Static Network with 300 Links

Photo BZ experiment

Successive snapshotsat 15 s intervals

Grid: 260 × 260 Cells: 10 × 10 grid pts26 × 26 = 676 cells

Threshold: 50% excited

Time Series of Fractional Coverage

(a) Experiments: 26 × 26 node network with 300 nonlocal links (red) and 20 nonlocal links (blue).

(b) Simulations: 50 × 50 node network with 200 nonlocal links (red) and 10 nonlocal links (blue).

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Time of Travel

Shading indicates time of travel from central node to a given node.

Weighted Shortest Distance Between Nodes

Local links Nonlocal links

Each local link is weighted as 1.Each nonlocal link is weighted as w.

We use a shortest path algorithm to calculate the distance between nodesin the network (Dijkstra’s Algorithm).

The weight w is chosen as the time necessary to initiate a nonlocal excitation in the BZ system. (w = nonlocal initiation time/local initiation time.) Typically w = 0.4.

C The distance between A and B is either4 (nonlocal links) or 2+ w.

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Shading of each node indicates theshortest distance between that node and the central node.

Shading of each node indicates timeof travel of BZ wave to that nodefrom the central node.

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Comparison of Network Distance and Time of Travel Model System: 50 x 50 nodes – 100 links

Distance Time of travel

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Shading of each node indicates theshortest distance between that node and the central node.

Shading of each node indicates timeof travel of BZ wave to that nodefrom the central node.

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Comparison of Network Distance and Time of Travel BZ Experiment: 26 x 26 nodes – 10 links

Distance Time of travel

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Time of travel vs shortest distance indicates that the excitation propagates through the network via the shortest path between nodes. (Related to determination of optimal paths in a maze, since a maze is equivalent to a network of connected nodes.)

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Comparison of Network Distance and Time of Travel Model System: 50 x 50 nodes – 100 links

Distance Time of travel

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Frequency Synchronization

50 x 50 node network – 25 nodes shown

Distributed Network Pacemakers

For few links: period of excitation loop is highly dependent on the network structure.

For > 50 links: period is governed by the period of the excitation cycle.

Error bars: standard deviation for 10 network configurations.

Excitation loops: links with destination nodes behind waves that excite the source nodes.

Period defined by the time required for a wave to travel from the destination node to the source node plus the time for the nonlocal link to re-excite the destination node.

Loop structure may have multiple local and nonlocal links.

Phase Synchronization of Nodes

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R = normalized vector sum of the phases

N = number of nodes

θj = phase of the jth node.

An R value close to zero indicates that the phases of the individual oscillators are evenly distributed, while an R value close to unity indicates the oscillators are in phase.

Active Links

Increasingly large spread in the values arises from increase in possible network configurations.

Pruning: a subset of links becomes inactive, while remaining active links account for dynamical behavior.

Active link: nonlocal link that results in successful wave initiation at destination cell.

Stability of Active Link Sequence

Link sequence unstable to perturbation (b) while stable to perturbation (c).

Sequence of active links: destination nodes are labeled according to the order of their excitations.

Sequence of initiations is repeated once per coverage oscillation.

Perturbation: (b) link 6 removed(c) link 4 removedfrom t = 24.4 to 40.0

Destination Node Locations of Active Links before and after Destabilizing Perturbation

Original perturbation: destination node of link 6 ● Lost: destination node of links 4 ● and 6 ● Gained: destination nodes of links 15-19 ●

BZ Experiment: Pruning of Links

Spontaneous shifting of active links during transient period as system relaxes to asymptotic sequence.

Network Architecture of Active Links

Mean path length: the average minimum separation of two nodes within the network.

Clustering coefficient: the average number of links between neighbors of an individual node divided by the total possible number of links between these neighbors.

Active links have same scaling as random links. However, only 1/3 to 1/4 of the random links are active. Dynamics with 50 active links is approx. same as with 150 random links. Pruning results in network optimization.

Dynamic Network

Successive snapshots from left to right.Grid: 1000 × 1000; Cells: 20 × 20 grid pts; Medium: 50 × 50 cellsJump threshold: 50% cell coverage; Jump probability: p = 0.02

Time Series and Power Spectrum

Jump probability p = 0.05

Fraction of excited elements

Time Series and Power Spectrum

Jump probability p = 0.4

Fraction of excited elements

Experimental Dynamic Link Network

Photo BZ experiment

Link probability p = 0.5

Successive snapshotsat 15 s intervals

Grid: 260 × 260 Cells: 10 × 10 grid pts26 × 26 = 676 cells

Threshold: 50% excited

Time Series and Power Spectra

Dynamic link network with p = 0.01 (dotted line) and p = 0.50 (solid line).

Coverage is the fraction of grid points in excited state. PSD is the power spectral density.

Domain Model Zipf Law for domain distribution.

Links to any domain (other than domain of origin) with a fixed probability to random location in any other domain.

Links from a domain proportional to departure domain size.

Links to a domain ca. proportional to destination domain size (for large populations).

Nearest neighbor interactions within a domain determine local spreading.

Grid: 1000 × 1000 Cells: 10 × 10 grid pts

Largest domain: 260 cells (population 2600)

Domains with Dynamic Links

Jump probability p = 0.1

Domains with Dynamic Links

Jump probability p = 0.3

Domains with Dynamic Links

Jump probability p = 0.5

Time Series for Excitation Coverage

p = 0.5

p = 0.3

p = 0.1

←Collapse!

Domains with Dynamic Links

Jump probability p = 0.5

Domains with Dynamic Links: Experiments

Jump probability p = 0.05

Domains with Dynamic Links: Experiments

Jump probability p = 0.7

Time Series for Excitation Coverage: Experimental

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H. Berry, and O. Temam, Lecture Notes in Computer Science, 3512:306-317, (2005)

Towards More Realistic Networks:Setting up the Links

Neurons do not connect with other neurons randomly over a spatial domain.

Neurons have a tendency to connect to nearby neurons and we approximate this tendency by an exponential falloff: P(d) = 1/ξ exp(-d/ξ) where P(d) is the connection probability at distance d, and ξ is the average connection distance parameter.

Nature of Links: Integrate and Fire

Neurons are activated according to an integrate and fire mechanism. In the context of the photo-BZ system, we have inputs to a node from all the connected nodes. The inputs are summed up and when they reach a threshold light intensity, the node fires (becomes excited). The destination node then becomes a source node with the excitation passed on according to the integrate and fire mechanism. We have assigned 20% of the links to be inhibitory.

Network Evolution without Diffusive Coupling

Evolution of network activity without diffusive coupling:1600 nodes and 125,000 links with 80% excitatory links (with weight of 0.03) and 20% inhibitory links (with weight of -0.08).Red represents excited state and blue represents steady state.

Network Evolution with Diffusive Coupling

Evolution of network activity with diffusive coupling:1600 nodes and 42,000 links with 80% excitatory links (with weight of 0.03) and 20% inhibitory links (with weight of -0.08).Red represents excited state and blue represents steady state.

Spike Timing-Dependent Plasticity (STDP)

• w represents the weight of a link, where • w → w + wp for potentiation.

• w → w + wd for depression.

pw STDPtpec

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M. Rossum, G. Bi & G. Turrigiano, “Stable Hebbian Learning from Spike Time-Dependent plasticity,” J. Neurosci. 20, 8812 (2000).

Application of STDP

Active links affect a target cell, wherewj is weight adjusted by changes wp or wd of link j based upon Δt.

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Link Weight Distribution

Distribution of link weights:1600 nodes and 125,000 links with 80% excitatory links (initial weight of 0.03) and 20% inhibitory links (not shown, constant weight of -0.08). Excitatory link weight increases from 0.03 to and average of ~0.06 by STDP mechanism.

Weighted Degree Distribution

Distribution of the sum of weighted links per node:Histogram shows number of nodes for each sum of weighted links.1600 nodes and 125,000 links with 80% excitatory links (with initial weight of 0.03) and 20% inhibitory links (constant weight of -0.08).

Spatiotemporal Networks in Addressable Excitable Media

Nearest neighbor spreading: Rxn-Diff wave

Non-diffusive spreading: fixed random links

Network distance and time of travel

Frequency and phase synchronization

Dynamic non-diffusive jumpsand domain model

Integrate and fire with inhibitory links

Spike-timing dependent plasticityNode

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