Macroscopic dynamics of neural networks with heterogeneous spiking thresholds

Open Access

Macroscopic dynamics of neural networks with heterogeneous spiking thresholds

Richard Gast, Sara A. Solla, and Ann Kennedy

Phys. Rev. E 107, 024306 – Published 16 February 2023

Abstract

Mean-field theory links the physiological properties of individual neurons to the emergent dynamics of neural population activity. These models provide an essential tool for studying brain function at different scales; however, for their application to neural populations on large scale, they need to account for differences between distinct neuron types. The Izhikevich single neuron model can account for a broad range of different neuron types and spiking patterns, thus rendering it an optimal candidate for a mean-field theoretic treatment of brain dynamics in heterogeneous networks. Here we derive the mean-field equations for networks of all-to-all coupled Izhikevich neurons with heterogeneous spiking thresholds. Using methods from bifurcation theory, we examine the conditions under which the mean-field theory accurately predicts the dynamics of the Izhikevich neuron network. To this end, we focus on three important features of the Izhikevich model that are subject here to simplifying assumptions: (i) spike-frequency adaptation, (ii) the spike reset conditions, and (iii) the distribution of single-cell spike thresholds across neurons. Our results indicate that, while the mean-field model is not an exact model of the Izhikevich network dynamics, it faithfully captures its different dynamic regimes and phase transitions. We thus present a mean-field model that can represent different neuron types and spiking dynamics. The model comprises biophysical state variables and parameters, incorporates realistic spike resetting conditions, and accounts for heterogeneity in neural spiking thresholds. These features allow for a broad applicability of the model as well as for a direct comparison to experimental data.

Received 9 September 2022
Accepted 2 February 2023

DOI:https://doi.org/10.1103/PhysRevE.107.024306

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Bifurcations Neuronal dynamics Synchronization transition

Collective dynamics

Nonlinear Dynamics

Authors & Affiliations

Richard Gast ^*, Sara A. Solla , and Ann Kennedy

Department of Neuroscience, Feinberg School of Medicine, Northwestern University, Chicago, Illinois 60611, USA

^*richard.gast@northwestern.edu

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 107, Iss. 2 — February 2023

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Strong spike-frequency adaptation reduces mean-field accuracy. (a) Bifurcations in the 2D parameter space spanned by the spike-frequency adaptation strength $κ$ and the input current $I$ . Green (gray) regions depict synchronized-oscillatory (asynchronous, bistable) regimes. Green (gray) solid lines depict curves of Hopf bifurcation (fold bifurcation) solutions. Orange squares, black diamonds, and green circles represent Bogdanov-Takens, cusp, and generalized Hopf bifurcations, respectively. Gray (green) $x$ markers depict fold (Hopf) bifurcations identified in the dynamics of a spiking neural network with neuron-specific recovery variables $u_{i}$ . (b), (c) Solution for the dimensionless mean-field variable $s$ as a function of $I$ for two different values of the spike frequency adaptation strength $κ$ . Purple solid lines represent stable steady-state solutions; gray dotted lines represent unstable steady-state solutions. Green solid lines represent the minima and maxima of limit cycle solutions. Gray triangles and purple circles mark fold and Andronov-Hopf bifurcations, respectively. Note that the fold bifurcation in (c) is the fold bifurcation of a limit cycle; it marks the collision of a stable and an unstable limit cycle branch in the close vicinity of the Andronov-Hopf bifurcation. This bifurcation landscape is typically found in the vicinity of generalized Hopf bifurcations [24]. (d), (e) Dynamics of the mean-field model (green), the spiking neural network with neuron-specific recovery variables $u_{i}$ (blue), and the spiking neural network with a global recovery variable $u$ (orange), for two different values of the spike frequency adaptation strength $κ$ . The top two rows depict the synaptic activation $s$ and recovery variable $u$ as a function of time, both averaged across neurons; the bottom row depicts the applied input current $I$ as a function of time. Note that for illustration purposes, input currents shown here increase much faster than the timescale used to identify approximate bifurcation points in (a), and that the input current in (d) was decreased (increased) in the second (first) half of the trial to locate the fold bifurcations that lie on the left (right) solution branch of the bistable region in (a). Vertical dashed lines in the plot for $u$ represent the troughs in $u$ used to locate fold bifurcation point (d) or the peaks in $u$ used to locate Hopf bifurcation points (e) for the input parameter $I$ . Horizontal dashed lines for $I$ in the bottom row of (d) and (e) indicate the values of $I$ where fold or Hopf bifurcation points occur, as identified from the troughs and peaks in $u$ , respectively.
Reuse & Permissions
Figure 2
Firing rates under different spike reset conditions. (a) Steady-state firing rates $r_{i}$ of single neurons as a function of the input $I$ for different reset potentials $v_{0}$ . Color code as in (b). (b) Differences between the steady-state firing rates of a neuron with $v_{0} \to - \infty$ as assumed in the mean-field model and neurons with different finite values of the reset potential $v_{0}$ . (c) Adjusted input $I^{*}$ as a function of $I$ for different reset potentials as per Eq. (36). Color code as in (b). (d) Steady-state solution for the dimensionless mean-field variable $s$ as a function of the input $I$ . Gray triangles represent fold bifurcations. Solid (dotted) lines represent stable (unstable) solutions. Feedback via the recovery variable $u$ [given by Eq. (40)] was turned off by choosing $κ = 0$ and $b = 0$ ; all other parameters were chosen according to Table 1. (e), (f) Synaptic activation dynamics $s$ for the spiking neural network, and for both uncorrected and corrected mean-field models, for two values of $v_{0}$ .
Reuse & Permissions
Figure 3
Bifurcation structure and network dynamics in an excitatory-inhibitory network for different spike reset conditions. (a)–(c) 2D bifurcation diagrams in the plane of fast-spiking spike threshold heterogeneity $Δ_{f s}$ and fast-spiking neuron input $I_{f s}$ for three different spike reset conditions. Regions of parameters space depicted in green (gray) represent synchronized-oscillatory (asynchronous-bistable) regimes as predicted by the corrected mean-field model. Solid green (gray) lines depict the Hopf (fold) curves predicted by the corrected mean-field model. Black diamonds represent cusp bifurcations, and green circles represent generalized Hopf bifurcations. Green (gray) $x$ markers depict the locations of Hopf (fold) bifurcations from the spiking neural network dynamics. (d)–(f) Same as (a)–(c), except that mean-field predictions follow from the uncorrected mean-field model. (g), (h) Synaptic activation variable $s$ (dimensionless) as a function of time for regular $rs$ (blue) and fast $fs$ (orange) spiking neurons, for two different spike reset conditions. Solid lines represent the solutions of the spiking network, dashed lines the mean-field model. The input to the fast-spiking neurons was stepped from $I_{f s} = 20 pA$ (no shading) to $I_{f s} = 30 pA$ (light gray shading), to $I_{f s} = 40 pA$ (dark gray shading). We chose $I_{r s} = 50 pA, Δ_{f s} = 0.3 mV$ , and all other parameters as specified in the Appendix.
Reuse & Permissions
Figure 4
Effects of a truncated Lorentzian distribution of spike thresholds on the mean-field dynamics. (a) Probability density as a function of the spiking threshold $v_{θ}$ for Lorentzian distributions with ${\bar{v}}_{θ} = - 40 mV$ and different widths $Δ_{v}$ . Both ${\bar{v}}_{θ}$ and $Δ_{v}$ are measured in mV. The transition from solid to dashed lines indicates the truncation. Dotted vertical lines represent two different values of the truncation threshold $ϕ$ , also measured in mV. (b), (c) Mean and variance of the difference $D_{r}$ between the average firing rates of the mean-field model vs the spiking network as a function of the Lorentzian width $Δ_{v}$ and the cutoff threshold $ϕ$ . (d), (e) Power-spectral density (PSD) of the average firing rate in the spiking network for different spiking thresholds and truncation thresholds, respectively. Both $Δ_{v}$ are $ϕ$ are measured in mV. We used $ϕ = 40 mV$ for (b) and (d) and $Δ_{v} = 1 mV$ for (c) and (e).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics