Deterministic excitable media under Poisson drive: Power law responses, spiral waves, and dynamic range

Tiago L. Ribeiro and Mauro Copelli

Phys. Rev. E 77, 051911 – Published 14 May 2008

Abstract

When each site of a spatially extended excitable medium is independently driven by a Poisson stimulus with rate $h$ , the interplay between creation and annihilation of excitable waves leads to an average activity $F$ . It has recently been suggested that in the low-stimulus regime $(h \sim 0)$ the response function $F (h)$ of hypercubic deterministic systems behaves as a power law, $F \sim h^{m}$ . Moreover, the response exponent $m$ has been predicted to depend only on the dimensionality $d$ of the lattice, $m = 1 / (1 + d)$ [T. Ohta and T. Yoshimura, Physica D 205, 189 (2005)]. In order to test this prediction, we study the response function of excitable lattices modeled by either coupled Morris-Lecar equations or Greenberg-Hastings cellular automata. We show that the prediction is verified in our model systems for $d = 1$ , 2, and 3, provided that a minimum set of conditions is satisfied. Under these conditions, the dynamic range—which measures the range of stimulus intensities that can be coded by the network activity—increases with the dimensionality $d$ of the network. The power law scenario breaks down, however, if the system can exhibit self-sustained activity (spiral waves). In this case, we recover a scenario that is common to probabilistic excitable media: as a function of the conductance coupling $G$ among the excitable elements, the dynamic range is maximized precisely at the critical value $G_{c}$ above which self-sustained activity becomes stable. We discuss the implications of these results in the context of neural coding.

Received 27 February 2008

DOI:https://doi.org/10.1103/PhysRevE.77.051911

Authors & Affiliations

Tiago L. Ribeiro^* and Mauro Copelli^†

Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil

^*tlr@df.ufpe.br
^†Corresponding author. mcopelli@df.ufpe.br

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 77, Iss. 5 — May 2008

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) (a) Membrane potentials of 100 ML neurons versus time with $h = 10^{- 2} {ms}^{- 1}$ for $G = 0.1$ (left panel), 0.2 (middle panel), and $0.3 mS / {cm}^{2}$ (right panel). The seed of the pseudo-random-number generator is the same for the three values of $G$ . (b) Response function for a one-dimensional lattice of $L = 1000 ML$ excitable elements. Symbols (bars) represent averages (standard deviations) over 10 runs. Solid lines are power laws discussed in the text.Reuse & Permissions
Figure 2
(Color online) (a) Snapshots of networks with $50 \times 50 ML$ neurons, with depolarized (spiking) membrane potentials coded as white. From left to right, $G = 0.1$ , $0.5,$ and $0.9 mS / {cm}^{2}$ , with $h = 2 \times 10^{- 3} {ms}^{- 1}$ and $t = 3.0 ms$ $(3.5 ms)$ for top (bottom) row. The seed of the pseudo-random-number generator is the same for the three values of $G$ . (b) Response function for a network of $200 \times 200 ML$ excitable elements. Symbols (bars) represent averages (standard deviations) over ten runs. Solid lines are power laws discussed in the text.Reuse & Permissions
Figure 3
Response curves of the GHCA model in $d = 3$ for increasing system size $(n = 3)$ . Inset: GHCA response function for $d = 2$ ( $L = 2744$ , averages over five runs). Dashed lines shows the response exponent $m = 1 / (1 + d)$ for both cases.Reuse & Permissions
Figure 4
(Color online) Spiral waves with $ϕ = 0.4 {ms}^{- 1}$ . (a) Snapshots of networks with $50 \times 50 ML$ neurons, with depolarized (spiking) membrane potentials coded as white. From left to right, $G = 0.15$ , $0.25$ , and $0.35 mS / {cm}^{2}$ , with $h = 4 \times 10^{- 3} {ms}^{- 1}$ and $t = 20 ms$ $(21 ms)$ for top (bottom) row. The seed of the pseudo-random-number generator is the same for the three values of $G$ . (b) Response function for a network of $200 \times 200 ML$ excitable elements. Symbols represent averages over five runs (standard deviations are smaller than symbol size). Firing rates were measured over 150 ms after a 50 ms transient. Horizontal and vertical lines illustrate the relevant quantities for calculating the dynamic range $Δ$ [see Eq. (7) and text for details].Reuse & Permissions
Figure 5
(Color online) Left (right) column: absence (emergence) of spiral waves. Dynamic range versus coupling conductance for $ϕ = 1 / 3 {ms}^{- 1}$ [(a) and (b)] and $ϕ = 0.4 {ms}^{- 1}$ (d). Triangles denote $G < G^{'}$ and squares denote $G^{'} < G < G^{″}$ . Circles denote $G > G^{″}$ in the absence of self-sustained activity (b), whereas pentagons denote the spiral wave regime (d). In (c), the self-sustained activity $F_{0}$ in the absence of stimulus is plotted against $G$ for fixed $ϕ = 0.4 {ms}^{- 1}$ . Inset of (c): estimated probability $p$ of spiral wave survival versus $G$ (see text for details). Inset of (d): response functions with $ϕ = 0.4 {ms}^{- 1}$ for $G = 0.25$ (solid squares) and $0.275 mS / {cm}^{2}$ (open pentagons). System sizes are $L = 1000$ in (a); $N = 200^{2}$ in (b) and (d); and $N = 150^{2}$ in (c).Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics