Griffiths phase and long-range correlations in a biologically motivated visual cortex model

Abstract

Activity in the brain propagates as waves of firing neurons, namely avalanches. These waves' size and duration distributions have been experimentally shown to display a stable power-law profile, long-range correlations and 1/f (b) power spectrum in vivo and in vitro. We study an avalanching biologically motivated model of mammals visual cortex and find an extended critical-like region - a Griffiths phase - characterized by divergent susceptibility and zero order parameter. This phase lies close to the expected experimental value of the excitatory postsynaptic potential in the cortex suggesting that critical be-havior may be found in the visual system. Avalanches are not perfectly power-law distributed, but it is possible to collapse the distributions and define a cutoff avalanche size that diverges as the network size is increased inside the critical region. The avalanches present long-range correlations and 1/f (b) power spectrum, matching experiments. The phase transition is analytically determined by a mean-field approximation.

Figure 1. Elements of the V1 model.

(A) Architecture of the network. (B) Spatial organization of the network of N = 4L neurons. The columnar structure is highlighted in red. There is a column of size N_c = 4l² = 196 neurons centered on each neuron of the network. (C) Compartmental scheme of neurons. The probability, , of choosing a presynaptic axonal compartment, a_k, of any neuron is exponential such that most of the synapses start from the end of the axon (left). The probability, , of choosing a dendritic postsynaptic compartment, d_m, is Gaussian with mean 50 and standard deviation 10, so that most of the synapses lay in the middle of the dendrite (right).

Figure 2. Network activity, order parameter and susceptibility.

Panel A: Temporal profile of the avalanches for many EPSP. Arrows mark the processing time T of the network. For E = 1.1 mV, only very small avalanches occur; E = 1.2 mV shows many small avalanches; for E = 2.0 mV there is a dominating avalanche (compare to the dotted line that marks zero activity); and for E = 13.0 mV the dominating avalanche takes over all the dynamics. Panel B: Density of activated neurons ρ during total activity time and its associated susceptibility χ_ρ = N(〈ρ²〉 − 〈ρ〉²)/〈ρ〉 as function of E for many L. Red circles (, , , ) indicate ρ and blue squares (, , , ) indicate χ_ρ; the larger L the darker the color shade. White background indicate the inactive phase, light gray background indicates the critical (Griffiths) phase with diverging χ_ρ and dark gray background indicates the percolating phase. Panels C,D: FSS of ρ order parameter yielding scaling exponent β/ν_⊥ = 0.55(3) – Equation (4) – and FSS of χ_ρ yielding scaling exponent γ/ν_⊥ = 3.1(1) – Equation (5); fits performed on the transition point E_c = 1.19 mV; () inactive phase, () critical phase, () and () percolating phases. Vertical bars are standard deviation and dotted lines are only guides to the eyes.

Figure 3. Avalanche size distributions and cumulative distributions.

Panel A: Typical avalanche distributions for many E: () inactive phase, () critical phase, () and () percolating phases. Note the bump around avalanches of size N_c ≈ 200 for E = 1.88 mV (); this bump reveals the internal structure of the network (see text for discussion). The light gray background highlights the range of avalanche size s in which all phases have PL-shaped distributions. Panel A inset: avalanche distributions for the critical phase (E = 1.15 mV) for increasing L. Panel B: avalanche cumulative distributions corresponding to panel A inset, E = 1.15 mV (the critical phase), for increasing L. Solid line is the fit of Equation (11) used to estimate Z(L = 20) and α = 1.4(1). Panel B inset: scaling law of Z ∼ L^D for E = 1.15 mV yielding D = 1.0(1); this scaling holds inside the whole critical phase. Avalanche distributions for the other phases have different D and are presented in Fig. 4.

Figure 4. Collapse of avalanche size cumulative distributions for many realizations of the simulation for each E.

Panel A: Collapse of avalanches in inactive phase; the cutoff does not scale with system size (D = 0) but the distribution presents a PL regime with exponent α = 1.5(1) due to noisy activity in the LGN. Panel B: Collapse of avalanches in Griffiths (critical) phase with exponents α = 1.4(1) and D = 1.0(1) corresponding to Fig. 3B. Panel C: Collapse of avalanches in weakly percolating phase with exponents α₁ = 1.4(1), α₂ = 1.72(8) and D = 3.1(3); the bump separating both PL ranges is a consequence of the columnar structure of the network, as it lies where s ≈ N_c. Panel D: Collapse of avalanches in strongly percolating phase with exponents α = 1.5(1) and D = 2; the gap in this distribution shows that propagation occurs mainly through a large dominating avalanche, so the PL scaling represents only noisy avalanche activity in the LGN. The straight black lines guide the eyes over the collapse of the PLs. Notice that α is the exponent of . All the PL exponents were checked using Maximum Likelihood test (see Supplementary Material).

Figure 5. Autocorrelation, power spectrum, DFA and processing time.

Panel A,B: Average autocorrelation (A) and average power spectrum (B) of avalanche sizes time series for L  = 99 and many E: () inactive phase, () critical phase, () and () percolating phases. The bold lines show the exponential cutoff fit of the autocorrelation for E = 1.6 mV giving τ = 16.2 ts (A); and the fit S(f) ∼ f^−b giving b = 0.97(5) for E = 1.15 mV (critical phase), b = 0.74(4) for E = 1.6 mV and b = 0.20(2) for E = 2 mV (weakly percolating phase) as examples (B). Notice how the curves’ slope in panel B smoothly vary and yield b ≈ 1 inside the GP (see panel C too). Panel C: Power spectrum and DFA exponents versus E inside and close to the GP. b is the avalanches power spectrum exponent and g is the DFA exponent of the activity time series (of Fig. 2A). Notice that b ≈ 1 and inside the critical region indicating long-range temporal correlations of both the avalanche time series and the spiking activity of the network. Panel D: Activity propagation time T as function of E for many L; solid line indicates the asymptotic behavior of T. Panel D inset: collapse plot yielding μ = 0.9(1). Arrows indicate the minimum of T, vertical bars are standard deviation, light gray is the GP and dark gray is the weakly percolating phase. See Supplementary Material for more details about T.