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Macroscopic description for networks of spiking neurons

Ernest Montbrió, Diego Pazó, Alex Roxin

arXiv:1506.06581v1q-bio.NCcond-mat.dis-nnnlin.AOnlin.CD

TL;DR

The paper addresses the lack of exact macroscopic descriptions linking firing rates to microscopic spiking-neuron states. It derives exact firing-rate equations and relates them through a conformal map to the Kuramoto order parameter.

  • Problem

    Existing firing-rate theories had not established an exact correspondence between network firing rates and the microscopic states of spiking neurons, limiting their applicability to synchronization.

  • Method

    The authors derive firing-rate equations for heterogeneous QIF-neuron networks and connect the Lorentzian and Ott–Antonsen representations through a conformal transformation.

  • Results

    The resulting ordinary differential equations are exact in the thermodynamic limit and reproduce network behavior far from fixed points under arbitrary external currents.

  • Takeaways & Limitations

    The equations provide a low-dimensional macroscopic description involving firing rate and mean membrane potential, including synchronous spiking states.

  • Takeaways & Limitations

    The derivation uses a quenched Lorentzian current distribution as a mathematical convenience, while broader distributions and independent noise are supported by additional analyses and simulations.

Abstract

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A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain function arises from the collective dynamics of networks of spiking neurons. This challenge has been chiefly addressed through large-scale numerical simulations. Alternatively, researchers have formulated mean-field theories to gain insight into macroscopic states of large neuronal networks in terms of the collective firing activity of the neurons, or the firing rate. However, these theories have not succeeded in establishing an exact correspondence between the firing rate of the network and the underlying microscopic state of the spiking neurons. This has largely constrained the range of applicability of such macroscopic descriptions, particularly when trying to describe neuronal synchronization. Here we provide the derivation of a set of exact macroscopic equations for a network of spiking neurons. Our results reveal that the spike generation mechanism of individual neurons introduces an effective coupling between two biophysically relevant macroscopic quantities, the firing rate and the mean membrane potential, which together govern the evolution of the neuronal network. The resulting equations exactly describe all possible macroscopic dynamical states of the network, including states of synchronous spiking activity. Finally we show that the firing rate description is related, via a conformal map, with a low-dimensional description in terms of the Kuramoto order parameter, called Ott-Antonsen theory. We anticipate our results will be an important tool in investigating how large networks of spiking neurons self-organize in time to process and encode information in the brain.

I. MODEL DESCRIPTION

The model describes heterogeneous, all-to-all coupled QIF neurons whose membrane potentials evolve under external and recurrent synaptic inputs with spike reset dynamics.

  • I. MODEL DESCRIPTION: QIF neurons provide the canonical model for Class I excitability near the spiking threshold.The firing-rate correspondence is exact in the thermodynamic limit, as N →∞.
  • I. MODEL DESCRIPTION: Each neuron emits a spike when its membrane potential reaches Vp, then resets to Vr.The analysis takes Vp = −Vr →∞; this rule captures spike reset and refractoriness.
  • I. MODEL DESCRIPTION: Input currents combine heterogeneous quenched inputs, a common time-varying external current, and recurrent drive proportional to synaptic activation.The recurrent contribution is the synaptic weight J multiplied by the mean synaptic activation s(t).
  • I. MODEL DESCRIPTION: Mean synaptic activation is constructed by summing normalized synaptic responses generated by presynaptic spikes.The single-spike response aτ(t) has time scale τ; an example is aτ(t) = e−t/τ/τ.

A. Continuous formulation

In the thermodynamic limit, the discrete neuron population becomes a conditional voltage density whose conservation is expressed by a continuity equation.

  • A. Continuous formulation: As N →∞, ρ(V |η, t)dV denotes the fraction of neurons with membrane potentials between V and V + dV at parameter η and time t.The heterogeneity parameter becomes a continuous random variable distributed according to g(η).
  • A. Continuous formulation: The total voltage density is obtained by integrating the conditional density over the heterogeneity distribution g(η).The formulation treats η as continuously distributed across the population.
  • A. Continuous formulation: Conservation of neuron number yields a continuity equation for the evolving voltage density.The equation explicitly includes the velocity specified by the microscopic neuron dynamics and input currents.

II. RESULTS

The unforced population has a Lorentzian stationary voltage distribution, and the Lorentzian Ansatz restricts the relevant macroscopic dynamics to a lower-dimensional manifold.

  • II. RESULTS: Without temporal forcing, the continuity equation has a stationary Lorentzian voltage density for each heterogeneity value η.For quiescent neurons, the density collapses at the rest state as a Dirac delta function.
  • II. RESULTS: The Lorentzian Ansatz assumes that solutions generically converge to a Lorentzian-shaped conditional density independently of initial conditions.This assumption places the relevant macroscopic dynamics inside a lower-dimensional space.
  • II. RESULTS: The Ansatz represents each conditional density with a time-dependent half-width x(η, t) and center y(η, t).The paper assumes this form completely describes the network’s macroscopic dynamics and later justifies its validity.

A. Macroscopic observables: Firing rate and mean membrane

The Lorentzian density directly connects its parameters to firing rate and mean membrane potential, providing macroscopic observables for analyzing steady states and bifurcations.

  • A. Macroscopic observables: Firing rate and mean membrane: The firing rate is computed as probability flux at infinity and has a simple relation to the Lorentzian half-width.The total firing rate r(t) is obtained by aggregating the rates across heterogeneity values.
  • A. Macroscopic observables: Firing rate and mean membrane: The phase diagram contains single low-activity stable nodes, single high-activity stable foci, and a bistable region with low- and high-rate states.The bistable region is bounded by a saddle-node bifurcation locus obtained exactly in parametric form.
  • A. Macroscopic observables: Firing rate and mean membrane: The system’s bifurcation diagrams track both firing rate r and mean membrane potential v across the mean external drive.Numerical QIF-network simulations provide the comparison points shown in the diagrams.
  • A. Macroscopic observables: Firing rate and mean membrane: The Lorentzian center y(η, t) equals the mean membrane potential for neurons with heterogeneity η.The mean is defined using a Cauchy principal value to avoid an otherwise ill-defined integral.

B. Firing-rate equations

The Lorentzian ansatz yields exact macroscopic equations for heterogeneous, all-to-all coupled QIF neurons in terms of firing rate and mean membrane potential, including strong synchrony.

  • The Lorentzian ansatz solves the membrane-potential continuity equation exactly and enables theoretical analysis of the network.
  • For infinitely fast synapses, synaptic activation equals the firing rate, reducing the network to equations for w(η,t).
  • A Lorentzian heterogeneity distribution reduces the description to two ordinary differential equations evaluated at η = ¯η − i∆.
  • The resulting equations are ˙r = ∆/π + 2rv and ˙v = v^2 + ¯η + Jr + I(t) − π^2r^2.
  • The nonlinear system describes population dynamics through firing rate r and mean membrane potential v.
  • Firing rate enters the membrane-potential equation through negative feedback, while coupling in the rate equation captures network-level effects of spike generation and reset.
  • For step and sinusoidal stimuli, the FREs exactly reproduce the QIF ensemble’s macroscopic transients up to finite-size effects.

C. Analysis of the firing-rate equations

Without forcing, the firing-rate equations have fixed-point attractors whose phase structure matches QIF-network simulations and includes damped macroscopic oscillations.

  • Without forcing, the system has only fixed-point attractors: a stable node, a stable focus, or bistability between low- and high-rate states.
  • The firing-rate equations show excellent correspondence with QIF-network bifurcation diagrams.
  • Stable-focus regions exhibit oscillatory decay, including around the high-activity state across much of the bistable region.
  • Damped macroscopic oscillations reflect transient synchronous firing by a fraction of neurons and are absent from traditional firing-rate models.

D. Analysis of the firing-rate equations: Non-stationary inputs

Under time-varying inputs, the firing-rate equations reproduce transient oscillations and bursting, and predict macroscopic chaos that also appears in QIF-network simulations.

  • The authors test step and sinusoidal stimuli by simulating both the full QIF network and the FREs.
  • A step current drives the system out of bistability toward a stable focus, and the FREs reproduce the network’s damped oscillations before and after current removal.
  • Periodic forcing across the bistable region produces periodic bursting when the system visits the stable-focus region.
  • Increasing the forcing frequency produces macroscopic chaos because the system cannot trivially follow the stable fixed point during each cycle.
  • The rate model and QIF simulations show similar chaotic attractors and irregular firing-rate dynamics.
  • Chaotic firing-rate maxima coincide with synchronous firing clusters of neurons having similar intrinsic currents, whose sizes vary irregularly over time.

E. Validity of the Lorentzian ansatz

The Lorentzian ansatz is connected to theta-neuron and Ott–Antonsen descriptions through a conformal map, while attraction to the relevant manifold remains incompletely proven for this system.

  • Transforming QIF voltage via V_j = tan(θ_j/2) converts the neurons into an ensemble of theta neurons.
  • The voltage and phase forms of the ansatz are Poisson-kernel representations related by a conformal map from the right half-plane to the unit disk.
  • The phase-space ansatz corresponds to the Ott–Antonsen manifold, which generally attracts dynamics in the thermodynamic limit.
  • For the QIF-derived system, convergence to the Ott–Antonsen manifold has been proven only when H is time-dependent but not η-dependent; attraction for η-dependent H is supported by theory and numerical studies.
  • The same map establishes a one-to-one relation between firing rate r, mean potential v, and the Kuramoto order parameter Z.

F. Firing rate and Kuramoto order parameter

The firing rate and mean membrane potential are related to the Kuramoto order parameter through a conformal mapping from the right half-plane to the unit disk.

  • F. Firing rate and Kuramoto order parameter: The macroscopic variables r and v are contained in W, while α measures the uniformity of the phase density.α = 0 corresponds to a perfectly uniform phase density.
  • F. Firing rate and Kuramoto order parameter: The Kuramoto order parameter Z is obtained by integrating α over the neuronal population.Z represents the center of mass of phases distributed around the unit circle.
  • F. Firing rate and Kuramoto order parameter: For Lorentzian current heterogeneity, an exact formula relates Z to the firing-rate quantity W ≡ πr + iv.The mapping is illustrated as a conformal transformation of the right half-plane r ≥ 0 onto the unit disk |Z| ≤ 1.

III. CONCLUSIONS

The paper presents exact firing-rate equations for heterogeneous QIF networks in the thermodynamic limit, while identifying assumptions and broader qualitative applicability beyond the Lorentzian setting.

  • III. CONCLUSIONS: The resulting ordinary differential equations are presented as the first exact firing-rate equations for a network of spiking neurons.The derivation applies to heterogeneous QIF neurons and is exact in the thermodynamic limit.
  • III. CONCLUSIONS: The derivation does not rely on weak coupling, time-scale separation, averaging, or other approximations.Validity assumes all-to-all connectivity, quenched heterogeneity, and Lorentzian-distributed inputs.
  • III. CONCLUSIONS: For arbitrary quenched heterogeneity, the Lorentzian ansatz remains generally valid for assessing network-state stability.Uniform and Gaussian inputs produce very similar macroscopic dynamics under time-varying inputs in numerical simulations.
  • III. CONCLUSIONS: The equations can be analyzed and simulated easily and exactly reproduce spiking-network behavior far from fixed points under arbitrary external currents.This contrasts with earlier derivations that mainly treated stationary states or weak deviations and often required specialized numerical methods.
  • III. CONCLUSIONS: The Ott-Antonsen description is related to the Lorentzian-ansatz description by a nonlinear transformation, while firing-rate equations provide a simple low-dimensional form.The firing-rate and mean-membrane-potential variables form a biophysically meaningful macroscopic observable corresponding to the Kuramoto order parameter.

APPENDIX B: PROOF OF EQUATION (18)

The appendix proves the conformal relation between the Kuramoto order parameter and the firing-rate field by transforming and evaluating a power-series representation.

  • APPENDIX B: PROOF OF EQUATION (18): The inverse of Eq. (18) is introduced before rewriting the macroscopic field W in terms of w ≡ x + iy.The proof recalls the definitions of the macroscopic quantities before applying the conformal transformation.
  • APPENDIX B: PROOF OF EQUATION (18): Substituting w = (1 − α)/(1 + α) transforms the expression into a series in powers of α.The transformation is the inverse of Eq. (15).
  • APPENDIX B: PROOF OF EQUATION (18): The resulting integrals are evaluated by grouping powers of α and identifying generalized order parameters Z_m.For Lorentzian g(η), the generalized parameters satisfy Z_m(t) = Z(t)^m, allowing the power series to be reverted to obtain Eq. (18).

APPENDIX C: RESULTS FOR ARBITRARY

For arbitrary current distributions, the Lorentzian-ansatz framework reproduces steady states, bifurcation boundaries, transient synchrony, bursting, and macroscopic chaos observed in QIF networks.

  • General distributions: The Lorentzian ansatz remains applicable to uniform and Gaussian current distributions, although these distributions do not satisfy the finite-pole condition.For distributions with 2n poles, the firing-rate equations consist of n complex-valued ordinary differential equations.
  • Steady states: Steady firing rates for arbitrary current distributions satisfy a self-consistency condition, agreeing with the firing-rate equations for Lorentzian distributions.The condition is obtained by integrating the firing rate over intrinsically active neurons.
  • Bifurcations: Saddle-node bifurcation loci can be computed systematically for arbitrary current distributions and agree with simulations for Gaussian distributions.For uniform distributions, the bifurcation branches emanate from a cusp at rescaled mean current −1/3.
  • Collective dynamics: The equations predict a stable focus whose trajectories show firing-rate and mean-potential oscillations caused by transient synchronous firing.This stable-focus behavior underlies the bursting and macroscopic chaos observed under periodic forcing.
  • Collective dynamics: Periodic forcing produces qualitatively similar bursting for uniform and Gaussian currents, while higher-frequency forcing yields an apparently chaotic state for Gaussian currents.The chaotic dynamics persists in the thermodynamic limit N →∞.
  • Independent noise: For identical QIF neurons driven by independent noise, firing-rate and mean-potential traces closely match the equations derived for quenched Lorentzian heterogeneity.The simulations also display transient spike synchrony and a near-quantitative fit to the firing-rate equations.
  • Model generalizations: The low-dimensional equations extend to distributed synaptic weights and currents, adding a coupling-heterogeneity term that shifts bistability toward lower mean currents and higher mean coupling.The generalized firing-rate equations include the additional term +Γr/π.
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