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Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size

Tilo Schwalger, Moritz Deger, Wulfram Gerstner

arXiv:1611.00294v3q-bio.NC

TL;DR

The paper addresses the unclear connection between single-neuron dynamics and large-scale population models. It derives stochastic mesoscopic equations from generalized integrate-and-fire networks and shows that they reproduce microscopic population statistics while capturing finite-size emergent dynamics.

  • Problem

    The connection between microscopic spiking neurons and population models remains largely unknown, despite its importance for describing rapid changes and fluctuations in neural activity.

  • Method

    The authors derive stochastic equations for interacting mesoscopic populations from randomly connected generalized integrate-and-fire neurons, incorporating refractoriness, adaptation, and finite population size.

  • Results

    The mesoscopic equations reproduce the temporal statistics of microscopic population activity and capture finite-size effects and nonlinear emergent dynamics.

  • Takeaways & Limitations

    The theory provides an efficient framework for replacing homogeneous-population spiking networks with interacting mesoscopic populations based on neuronal and synaptic parameters.

  • Takeaways & Limitations

    The framework assumes homogeneous populations, although weak heterogeneity is allowed and strongly heterogeneous groups may require subdivision.

Abstract

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Neural population equations such as neural mass or field models are widely used to study brain activity on a large scale. However, the relation of these models to the properties of single neurons is unclear. Here we derive an equation for several interacting populations at the mesoscopic scale starting from a microscopic model of randomly connected generalized integrate-and-fire neuron models. Each population consists of 50 -- 2000 neurons of the same type but different populations account for different neuron types. The stochastic population equations that we find reveal how spike-history effects in single-neuron dynamics such as refractoriness and adaptation interact with finite-size fluctuations on the population level. Efficient integration of the stochastic mesoscopic equations reproduces the statistical behavior of the population activities obtained from microscopic simulations of a full spiking neural network model. The theory describes nonlinear emergent dynamics like finite-size-induced stochastic transitions in multistable networks and synchronization in balanced networks of excitatory and inhibitory neurons. The mesoscopic equations are employed to rapidly simulate a model of a local cortical microcircuit consisting of eight neuron types. Our theory establishes a general framework for modeling finite-size neural population dynamics based on single cell and synapse parameters and offers an efficient approach to analyzing cortical circuits and computations.

Author Summary

The paper develops a mesoscopic link between microscopic spiking neurons and coarse neural population activity. Its stochastic equations account for finite-neuron fluctuations, reproduce microscopic simulations, and support efficient multiscale modeling.

  • The study derives stochastic population equations at the mesoscopic scale of 100–1000 neurons from an underlying microscopic model.The equations are designed to connect microscopic and macroscopic activity patterns.
  • The equations account for fluctuations caused by the finite number of neurons.
  • Mesoscopic simulations reproduce microscopic simulation results while achieving a high speed-up factor.

Introduction

The introduction identifies a missing quantitative connection between single-neuron physiology and finite-size population models. It proposes deriving non-perturbative stochastic equations from generalized integrate-and-fire networks to support efficient, experimentally grounded cortical modeling.

  • Limitations of existing models: Existing neural mass models compactly summarize activity but do not correctly reproduce rapid responses, realistic fluctuations, or links to local field potentials.
  • Approach: The paper derives interacting mesoscopic population equations from generalized integrate-and-fire neuron models with parameters that can be extracted from experiments.
  • Motivation: Mesoscopic models target cortical populations of 50–2000 neurons, where finite-size fluctuations remain important.
  • Aims and validation: The framework aims to predict how neuronal parameters or inputs affect interacting populations and to enable efficient coarse-grained multiscale simulations.
  • Finite-size and history effects: Spike-history effects such as refractoriness and adaptation make population activity non-Poissonian and generate temporally correlated synaptic input.
  • Novelty: The approach consistently incorporates strong refractoriness, finite population size, small populations, non-Gaussian fluctuations, and dynamics far from stationary states.

Results

The derived mesoscopic equations reproduce microscopic population statistics while retaining finite-size fluctuations and spike-history effects. They capture refractoriness, adaptation, burstiness, synchronization, and stochastic switching across network settings.

  • Finite-size effects: Finite-size fluctuations are quantitatively relevant because experimentally estimated local cortical populations contain relatively few neurons.The framework targets populations of finite size rather than assuming fluctuations vanish in a large-population limit.
  • Mesoscopic population equations: The coupled mesoscopic equations provide the paper’s main result, with Gaussian noise summarizing microscopic stochasticity while microscopic state detail is removed.The dynamics is determined by the history of mesoscopic population activity.
  • Comparison of microscopic and mesoscopic simulations: Mesoscopic simulations reproduce microscopic temporal statistics for uncoupled neurons, including expected activity trajectories and refractory power-spectrum structure.The agreement persists at high firing rates, where refractoriness produces a low-frequency power dip.
  • Spike-history effects: The equations capture adaptation and burstiness qualitatively, including burst-related low-frequency power and adaptation-related suppression at still lower frequencies.The quasi-renewal approximation systematically overestimates power across frequencies in the bursty example.
  • Recurrent networks: In randomly connected E-I networks, shared finite-size fluctuations produce correlated, partially synchronized activity that decreases as network size increases.Microscopic and mesoscopic power spectra agree even for an extremely small network of one inhibitory and four excitatory neurons.

Discussion

The study derives stochastic mesoscopic population equations from microscopic spiking neurons, linking single-cell history effects and finite-size fluctuations to cortical population dynamics. The framework reproduces microscopic network activity, supports efficient simulation and parameter inference, and extends toward neural fields and multiscale applications.

  • Quantitative modeling of mesoscopic neural data: applications and experimental predictions: The theory replaces homogeneous-population spiking networks with interacting mesoscopic populations while allowing weak heterogeneity and subdivision by neuronal type.It is expected to break down when a few outlier neurons dominate activity, with further limits from mean-field and quasi-renewal approximations.
  • Quantitative modeling of mesoscopic neural data: applications and experimental predictions: The equations correctly predict both means and fluctuations in a detailed cortical microcircuit simulation while capturing finite-size-induced effects on network dynamics.They extend population-density approaches by predicting fluctuations and their effects, including stochastic oscillations.
  • Theoretical aspects: The framework generalizes deterministic infinite-population equations to stochastic finite-population dynamics without relying on perturbation around the infinite-size limit.This enables representation of large and non-Gaussian fluctuations in strongly nonlinear systems such as bistable networks.
  • Theoretical aspects: Unlike phenomenological stochastic rate models, the noise is derived from microscopic dynamics without a free intensity parameter and incorporates refractoriness through history-dependent effective fluctuations.Earlier models commonly add ad hoc white noise to deterministic rate dynamics and neglect strong negative self-feedback from refractoriness.
  • Extensions of the model: The equations provide a basis for low-dimensional dynamics, stochastic neural-field models, multiscale brain simulations, and efficient inference of unknown microscopic or synaptic parameters.Their spatial-continuum extension remains a proposed direction for studying spike-history effects in neural-field bumps and waves.
  • Theoretical aspects: Mesoscopic population dynamics achieved a speed-up of around 120 over microscopic simulation, with potentially larger gains from larger integration steps.A moderate performance enhancement was also observed for sufficiently large networks, N ≳100.

Model

The model describes interacting populations of generalized integrate-and-fire neurons with synaptic coupling, spike-history effects, and stochastic spike emission. Population-specific parameters represent neuron and synapse properties, including refractoriness, adaptation, and threshold softness.

  • Network architecture: Each population contains randomly connected neurons of the same type, with fixed in-degree connections between populations.The model uses population labels for neuron types and connection probabilities between populations.
  • Synaptic coupling: Synaptic input is constructed from delayed presynaptic spikes filtered by normalized postsynaptic-current kernels and weighted by population-specific strengths.Synaptic kernels can be represented using a finite set of synaptic variables for efficient simulation.
  • Neuron dynamics: Neurons follow generalized integrate-and-fire dynamics with membrane reset, absolute refractoriness, dynamic thresholds, and escape-noise spike generation.After a spike, voltage is reset and clamped for 4 ms; spikes are emitted stochastically through a conditional intensity.
  • Neuron dynamics: The spike-triggered threshold kernel accumulates across spikes to model adaptation and is set to infinity during the refractory period.This formulation captures both spike-frequency adaptation and absolute refractoriness through threshold dynamics.
  • Spike generation: The escape rate depends exponentially on membrane-potential distance from threshold, with threshold softness controlling intrinsic stochasticity.As threshold softness approaches zero, the model approaches a deterministic adaptive integrate-and-fire neuron.
  • Model variant: The generalized linear-model variant replaces voltage reset with spike-triggered membrane-potential kernels, but this mapping is not exact because firing-voltage variability and threshold accumulation remain.The conditional firing rate is invariant under a transformation between voltage and threshold kernels, although membrane potentials are not preserved exactly.

Mean-field approximation

The mean-field reduction replaces neuron-specific synaptic input with population activity and summarizes spike-history effects using each neuron’s last spike time. A quasi-renewal approximation then yields a tractable conditional firing rate for the population description.

  • Population state: The reduction tracks each neuron’s last spike time because it predicts refractory state and mediates coupling and adaptation in the population-density description.The resulting density describes the distribution of refractory states across neurons.
  • Mean-field synaptic input: For fully connected populations, all neurons receive identical mean-field synaptic input, making membrane potential a function of time and last spike time.In this limit, the mean-field representation follows directly from the network dynamics.
  • Mean-field synaptic input: For randomly connected fixed-in-degree networks, population averaging separates coherent activity fluctuations from neuron-specific deviations treated as incoherent noise.The approximation can remain reasonable for relatively sparse networks, including connection probability p = 5%.
  • Finite-size fluctuations: Finite populations retain common activity fluctuations because incoherent noise averages out as population size grows, while nonlinear averaging can soften the effective escape threshold.This identifies the finite-size component that survives the mean-field reduction.
  • Quasi-renewal approximation: The quasi-renewal approximation averages earlier spike history as an inhomogeneous Poisson process, reducing the firing rate’s dependence to the last firing time.The resulting approximation no longer requires the precise spike history of each neuron and its presynaptic partners.

Discretized population density equations.

The discretized population-density formulation tracks how neurons move between last-spike-time cohorts and generates stochastic population spike counts. Its continuum limit produces a Langevin equation for refractory density with Gaussian finite-size noise.

  • Refractory distribution: The refractory distribution m(t_l,t_k) records neurons whose last spike occurred in interval t_k, and its normalized form fully characterizes population refractory state.The same quantity can be interpreted either as a distribution of last spike times or as a survival number.
  • Cohort dynamics: Each cohort’s survival number decreases when its neurons fire again, with the update governed by independent conditional spike counts.Absolute refractoriness sets the initial cohort size and prevents more than one spike per neuron within a sufficiently short time step.
  • Computational reduction: The microscopic population-density simulation requires one random draw per past time interval and approaches the complexity of simulating all neurons as the time step vanishes.This motivates the later coarse-grained mesoscopic formulation.
  • Continuum limit: In the continuum and large-population limit, refractory density follows a Langevin equation driven by a Gaussian random field with delta-correlated time and last-spike-time coordinates.The population activity is obtained by integrating changes in refractory density across all refractory states.
  • Finite-size statistics: The refractory-density variance is zero immediately after firing and at long times, producing a maximum at an intermediate time.The intermediate maximum reflects the competing effects of initially complete survival and eventual cohort depletion.
  • Mesoscopic update: The mesoscopic discrete-time equation computes the present mean spike count from past spike counts, then samples the current count as a Poisson random variable.This provides the stochastic update rule for population activity at sufficiently small time steps.

Mesoscopic population density equations in continuous time

The continuous-time formulation represents population activity as a history-dependent point process coupled to expected activity and finite-size fluctuations. Its Gaussian approximation exposes multiplicative noise whose magnitude decreases with population size while retaining correlated spike-history effects.

  • Continuous-time population dynamics: Population activity is a history-dependent point process with conditional intensity N ¯A(t), rather than an inhomogeneous Poisson process with an externally prescribed rate.The expected activity depends on past population activity, so each spike-history realization changes the conditional intensity.
  • Continuous-time population dynamics: The coupled equations make ¯A(t) and AN(t) mutually dependent through feedback between expected activity and realized population spikes.The expected activity is determined by past activity, while realized activity feeds back into its dynamics.
  • Gaussian approximation: For large N, conditionally independent spike counts yield a Gaussian-process approximation with state-independent white noise entering the population activity multiplicatively.The approximation follows from spike-count means and variances N ¯A(tl)∆t and makes the multiplicative noise explicit.
  • Gaussian approximation: Finite-size fluctuations scale like 1/N, while population activity combines a white-noise spike-train term with a correlated colored component generated by filtering through the activity dynamics.The resulting autocorrelation contains both a δ-peak and a continuous part.
  • Spike-history effects: Refractoriness can produce negative short-lag autocorrelations because positive white-noise fluctuations are anti-correlated with the filtered expected activity.This connects spike-history effects at the single-neuron level to observable population-level correlations.

Several populations

The framework extends to multiple interacting populations by assigning population labels and history-dependent expected activities. Its full formulation contains age- or last-spike-time-indexed dynamics coupled through population interactions.

  • Several populations: Multiple populations are incorporated by adding a population label α = 1, . . . , M to the population equations.Each population has its own activity while the equations retain the shared stochastic framework.
  • Several populations: The expected activity of each population depends explicitly on the history of activities across populations, yielding interacting population dynamics.The conditional intensity is determined by the population’s expected activity and its history.
  • Several populations: The full equations comprise ordinary differential equations for variables S, u, and v indexed by the continuous last-spike time, together with five integrals over that history variable.The paper subsequently gives equivalent partial-differential and finite-history formulations.

Refractory density representation

An age-density representation reformulates the last-spike-time equations using τ = t − ˆt, paralleling density-based descriptions of neuronal states. The resulting equations determine expected activity and evolve age, adaptation, and synaptic variables with boundary conditions.

  • Refractory density representation: The equivalent formulation uses neuronal age τ = t − ˆt as the state variable, analogous to membrane-potential density formalisms.This representation replaces explicit last-spike-time indexing with an age coordinate.
  • Refractory density representation: The age density q(t, τ) is defined from survival probability, past activity, and age, but finite-size fluctuations mean it is not normalized as a probability density.The density is q(t, τ) = S(t|t −τ)AN(t).
  • Refractory density representation: The density equation yields expected population rate from the age distribution, analogous to computing probability flux from a membrane-potential density.The stochastic activity is then obtained from the population activity equation.
  • Refractory density representation: The auxiliary variables u and v satisfy corresponding first-order partial differential equations with boundary conditions u(t, 0) = ur and v(t, 0) = 0.Together with threshold dynamics, these equations provide the age-based population description.
  • Refractory density representation: The earlier ordinary differential equations are the characteristic equations of the age-density partial differential equations.This establishes equivalence between the last-spike-time and age-based representations.

Population equations for a finite history.

The finite-history formulation replaces integrals over the infinite past with explicit treatment of recent refractory history and compact variables for older free neurons. The history length is chosen from refractory and adaptation-kernel conditions, enabling a finite-dimensional implementation.

  • Population equations for a finite history: The infinite history is truncated to a window t − T ≤ ˆt < t, while older neurons are treated separately using an average threshold based on a Poisson-history assumption.This split distinguishes recent refractory neurons from older free neurons whose last-spike effects have decayed.
  • Population equations for a finite history: The history length T is determined mainly by refractory time scales and by conditions ensuring that the quasi-renewal kernel approximates the spike-triggered kernel.The required conditions need not make T larger than the largest kernel time scale.
  • Population equations for a finite history: For an exponential adaptation kernel, the finite-history condition can be met by sufficiently small adaptation time scale or by sufficiently large time scale satisfying Jθ/τθ ≪ ∆u.The adaptation strength also determines the firing-rate reduction in the strong-drive limit.
  • Population equations for a finite history: The finite-history equations retain exact recent refractory effects while approximating older spike-triggered effects through exponential-kernel variables and free-neuron threshold dynamics.Free and refractory thresholds are then used to obtain their respective conditional intensities.
  • Population equations for a finite history: The history integrals are reduced to the finite interval [t − T, t), while long past tails are represented by differential equations for compact variables such as the expected number of free neurons.This avoids directly integrating over the entire past during simulation.

Numerical implementation

The implementation discretizes time and updates membrane potentials, thresholds, firing probabilities, population variables, and finite spike histories while enforcing timestep conditions for numerical validity.

  • Numerical implementation: The timestep must be small enough that neurons fire at most once per step, does not exceed transmission delay, and satisfies additional approximation conditions.The expected spike count is also assumed small relative to population size for the Poisson-statistics derivation, although efficient integration may benefit from larger steps.
  • Numerical implementation: The algorithm updates membrane potentials and thresholds separately for free and refractory neurons, maintaining reset values during the absolute refractory period.Refractory histories use a finite number K of bins, with the oldest and most recent spike times tracked explicitly.
  • Numerical implementation: Firing probabilities are obtained from discretized intensities, with conditional refractory intensities set to zero during the absolute refractory period.Free and conditional probabilities use the form 1 − e^(-λ̄∆t).
  • Numerical implementation: Population dynamics use discrete updates for survival-number means and variances, including the finite-history variables x and z and their initial conditions.The variables m̄_k and v_k represent the mean and variance of unconstrained survival numbers.
  • Numerical implementation: Each update computes the expected spike count and samples the actual count from a binomial distribution over the population.The binomial distribution represents N Bernoulli trials with the relevant success probability.
  • Numerical implementation: The implementation can initialize all neurons synchronously in a refractory state and stores history in circular buffers for population updates.The update procedure returns the spike count and updated state variables at the next timestep.

Power spectrum

The power-spectrum analysis characterizes stationary population fluctuations using Fourier transforms and relates them to renewal-neuron theory. For finite populations, fluctuation power decreases with population size and vanishes in the macroscopic limit.

  • Power spectrum: Stationary population activity fluctuations are characterized through the power spectrum of activity measured over a time window T.The population activity is transformed into frequency space for spectral analysis.
  • Power spectrum: For renewal neurons, the population power spectrum is available analytically from the Fourier transform of the interspike-interval density and the stationary firing rate.The renewal-neuron expression provides the theoretical reference for fluctuation spectra.
  • Power spectrum: Fluctuation power scales as 1/N and vanishes as N →∞ for the LIF model with escape noise.The analysis uses the hazard rate and the stationary firing rate to connect population activity spectra with renewal-neuron results.

Modified Potjans-Diesmann model

The modified Potjans-Diesmann model adapts the original cortical-column parameters for the mesoscopic framework while matching its spontaneous stationary firing rates.

  • Modified Potjans-Diesmann model: The model replaces background Poisson input with constant drive and increased escape noise, while fitting GIF resting potentials to roughly match target firing rates.The modified parameters include ur = 0 mV, uth = 15 mV, and ∆u = 5 mV.
  • Modified Potjans-Diesmann model: The mean current drive is chosen so spontaneous firing rates match those of the original model, with the full parameter set reported in Methods.The population equations enable efficient evaluation of firing rates during this fitting procedure.
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