Source-linked AI summary
Dynamical synapses causing self-organized criticality in neural networks
Anna Levina, J. Michael Herrmann, Theo Geisel
TL;DR
The paper studies how dynamical synapses generate self-organized criticality in spiking neural networks. It derives mean-field quantities and shows that critical avalanche dynamics arise broadly, becoming critical for any α ≥ 1 in the thermodynamic limit.
Problem
The paper asks whether synaptic dynamics alone can produce self-organized critical avalanche activity in spiking neural networks.
Method
The paper derives analytical mean-field expressions for synaptic coupling and inter-spike dynamics in spiking networks with dynamical synapses.
Results
The network exhibits approximate power-law avalanche distributions, and in the thermodynamic limit becomes critical for any α ≥ 1.
Takeaways & Limitations
Depressing synapses are sufficient to evoke self-organized critical avalanches; facilitating synapses are not necessary.
Takeaways & Limitations
The analysis focuses on fully connected networks and neurons without leak currents, while assuming external input is absent during avalanches under τ_d ≪ τ.
Abstract
from arXiv · showhide
We show that a network of spiking neurons exhibits robust self-organized criticality if the synaptic efficacies follow realistic dynamics. Deriving analytical expressions for the average coupling strengths and inter-spike intervals, we demonstrate that networks with dynamical synapses exhibit critical avalanche dynamics for a wide range of interaction parameters. We prove that in the thermodynamical limit the network becomes critical for all large enough coupling parameters. We thereby explain experimental observations in which cortical neurons show avalanche activity with the total intensity of firing events being distributed as a power-law.
sp) −θ
Activity-dependent synaptic efficacy produces a feedback between neurotransmitter depletion, recovery, and network firing. This dynamics supports avalanche behavior ranging from subcritical to approximately power-law regimes.
- Synaptic dynamics: Synaptic strength Jij represents available neurotransmitter and decreases by fraction u when a presynaptic spike arrives.During silence, the synapse recovers toward α/u on the slow timescale τJ.
- Avalanche regimes: Small α yields subcritical avalanche-size distributions with very few avalanches, while intermediate α produces approximate power laws extending nearly to system size.At larger α, the distribution becomes non-monotonous.
- Avalanche regimes: Depressing synapses alone are sufficient to evoke self-organized critical avalanches in spiking neural networks.This agrees with observations that synapses connecting excitatory neurons are largely depressive.
- Avalanche regimes: Matching a static network’s uniform strength to α0 = u⟨Jij⟩ reproduces the dynamical network’s power-law exponent and mean-squared deviation.Thus, average synaptic strength is sufficient to identify the avalanche-size distribution in the comparison described here.
- Synaptic dynamics: Short inter-spike intervals limit recovery and lower average synaptic strength, whereas stronger synapses produce longer avalanches that shorten subsequent intervals.These opposing effects create a usage-dependent feedback between synaptic efficacy and firing times.
and α
The paper derives a self-consistency relation linking average synaptic strength and inter-spike intervals, then validates its stationary solution against simulations. Near critical strength, the solution varies weakly with α, producing a broad critical region.
- Self-consistency analysis: The graphical solution agrees well with simulations for α = 1.3, ..., 2.0 in steps of 0.1.The comparison uses networks with N = 500, ν = 10, and u = 0.2.
- Self-consistency analysis: Analytical stationarity equations determine the mutually dependent averages ⟨Jij⟩ and ⟨∆isi⟩.Their graphical solution is given by intersections of the corresponding equations.
- Critical-region stability: The stationary solution is unique for each α and less sensitive to α near the critical synaptic strength.This reduced sensitivity accounts for the large critical region observed with depressive synapses.
- Critical-region stability: In simulations, a few spikes drive the average synaptic strength back to the fixed point, stabilizing the critical state.This supports stability of the self-consistency solution.
- Thermodynamic scaling: The analysis and numerical study initially concern finite systems, motivating examination of how the critical parameter interval changes with network size.The reported deviation threshold for this examination is ∆γ = 0.005.
and ǫ > w
The thermodynamic analysis distinguishes scaling regimes for the mean inter-spike interval and determines when self-consistent solutions can exist. Different exponents impose different constraints on α and the limiting average efficacy.
- Scaling regimes: When ⟨∆isi⟩ scales as N^(1+ε), the self-consistency right-hand side tends to −∞, precluding a solution under the stated conditions.
- Scaling regimes: When ⟨∆isi⟩ scales as N^(1+ε), a solution is possible only if α = 1, with u⟨Jij⟩ tending to 1.
- Scaling regimes: When ⟨∆isi⟩ scales as N^(1−ε), the self-consistency right-hand side tends to 1 while the left-hand side approaches α.
and ǫ < 0
In the thermodynamic limit, dynamical synapses produce criticality for all α ≥ 1, and the critical behavior remains robust across several network and neuron generalizations examined numerically.
- Thermodynamic limit: For α > 1, the thermodynamic analysis gives a unique fully connected network scaling with ⟨∆isi⟩ = cN and u⟨Jij⟩ tending to 1.The stated solution is c = −ν[ln(α − 1) − ln(α − 1 + u)].
- Scope: The study focuses analytically on fully connected networks and neurons without leak currents for tractability.
- Generalizations: Partially connected random networks retain the results when synaptic strengths are properly rescaled.The paper also reports more accurate power laws for small-world connectivity and grid networks than for independent random connectivity with the same average connectivity.
- Generalizations: With a biologically realistic leak term, avalanche sizes remain power-law distributed for leak time constants up to τl ≈ 40 ms.A compensatory synaptic current is included to preserve near-threshold firing probability.
- Thermodynamic limit: In the thermodynamic limit, networks with dynamical synapses become critical for any α ≥ 1.The paper identifies this criticality as intrinsic to synaptic dynamics.
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The paper develops a mean-field and self-consistency analysis linking dynamical synaptic strengths, avalanche statistics, and inter-spike intervals. It uses these relations to characterize critical behavior and its dependence on synaptic dynamics and interaction parameters.
- Analysis: The average synaptic strength is treated as an expectation over the synaptic dynamics and is related primarily to the mean inter-spike interval.The analysis assumes that ⟨Jij⟩ depends essentially only on the mean inter-spike interval.
- Analysis: The model connects avalanche and inter-avalanche interval distributions through mean-field relations and external inputs accumulated until threshold.The mean avalanche size ⟨L⟩ enters the relation between inter-spike and inter-avalanche intervals.
- Analysis: For dynamical synapses, the static-synapse strength is replaced by α0 = u⟨Jij⟩ to compute the mean avalanche size as a function of average coupling.This substitution allows the analytical avalanche calculation for static synapses to be applied under a mean-field approximation.
- Analysis: The self-consistency equations combine the inter-spike and synaptic-strength relations, with their solution obtained by numerical analysis.The equations arise from the two independent relations and are used to determine the coupled dynamical state.