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Phase response approaches to neural activity models with distributed delay

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Abstract

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson–Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.

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Acknowledgements

This work was supported by ANR-Ermundy and the Basic Science Program of the NRU Higher School of Economics.

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Authors and Affiliations

  1. Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, 24 rue Lhomond, 75005, Paris, France

    Marius Winkler, Grégory Dumont & Boris Gutkin

  2. Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623, Berlin, Germany

    Marius Winkler & Eckehard Schöll

  3. Center for Cognition and Decision Making, Institue for Cognitive Neuroscience, NRU Higher School of Economics, Krivokolenniy sidewalk 3, 101000, Moscow, Russia

    Boris Gutkin

  4. Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität, Philippstraße 13, 10115, Berlin, Germany

    Eckehard Schöll

  5. Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473, Potsdam, Germany

    Eckehard Schöll

Authors
  1. Marius Winkler
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  2. Grégory Dumont
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  3. Eckehard Schöll
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  4. Boris Gutkin
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Corresponding author

Correspondence to Boris Gutkin.

Additional information

Communicated by James Maclaurin.

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This article belongs to S.I. : Stochastic Oscillators.

Appendices

Appendix A

1.1 Derivation of the adjoint equation

To derive the adjoint equation for dynamics with distributed delay with the phase response curve Z(t) and the perturbed linearized variable \({X_p}(t)\), we must show that:

$$\begin{aligned} \left\langle {Z}(t), {X_p}(t)\right\rangle {\mathop {=}\limits ^{!}} ~\mathrm {const.} \end{aligned}$$

where \({Z}(t) \in \mathbb {R}^{N}\) and \({X_p}(t) \in \mathbb {R}^{N}\) are row vectors of N real components.

We prove this by showing that:

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} \left\langle {Z}(t), {X_p}(t)\right\rangle {\mathop {=}\limits ^{!}}&~0 \end{aligned}$$
(14)

with the scalar product defined as

$$\begin{aligned}&\left\langle {\Psi }(t), {\Phi }(t)\right\rangle = {\Psi }(t) {\Phi }(t) \nonumber \\&\quad + \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) \int _{t-s}^{t} \mathrm{d}\xi ~~ {\Psi }(\xi +s) ~ {DF}_2(\xi +s) ~ {\Phi }(\xi ) \end{aligned}$$
(15)

where \({DF}_2(t)\) is the Jacobian with respect to the delayed term of the perturbed linearized equation \( \frac{\mathrm {d}}{\mathrm {d}t}{X_p}(t) = {DF}_1(t){X_p}(t) + {DF}_2(t) \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) {X_p}(t-s) \). We apply the scalar product to Eq. (14) and obtain:

$$\begin{aligned}&\Leftrightarrow \frac{\mathrm {d}}{\mathrm {d}t} \Biggl [{Z}(t) {X_p}(t) + \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) \int _{t-s}^{t} \mathrm{d}\xi ~~ {Z}(\xi +s) ~ {F}_2(\xi +s) ~ {P}(\xi )\Biggr ]&= ~0 \end{aligned}$$

We obtain:

$$\begin{aligned}&\Leftrightarrow \Biggl (\frac{\mathrm {d}}{\mathrm {d}t} {Z}(t)\Biggr ) {X_p}(t) \\&\quad + {Z}(t) \frac{\mathrm {d}}{\mathrm {d}t}{X_p}(t) \\&\quad + \frac{\mathrm {d}}{\mathrm {d}t} \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) \int _{t-s}^{t} \mathrm{d}\xi ~~ {Z}(\xi +s) ~ {DF}_2(\xi +s) ~ {X_p}(\xi )=~ 0\\ \end{aligned}$$

Taking the derivatives of the integrals:

$$\begin{aligned}&\Leftrightarrow \Biggl (\frac{\mathrm {d}}{\mathrm {d}t} {Z}(t)\Biggr ) {X_p}(t) \\&\quad + {Z}(t) \frac{\mathrm {d}}{\mathrm {d}t}{X_p}(t) \\&\quad + \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s)~ {Z}(t+s) ~ {DF}_2(t+s) ~ {X_p}(t) \\&\quad - \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s)~ {Z}(t) ~ {DF}_2(t) ~ {X_p}(t-s)=~ 0\\ \end{aligned}$$

We apply \(\frac{\mathrm {d}}{\mathrm {d}t}{X_p}(t) = {DF}_1(t){X_p}(t) + {DF}_2(t) \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) {X_p}(t-s)\) and simplify the integral:

$$\begin{aligned}&\Leftrightarrow \Biggl (\frac{\mathrm {d}}{\mathrm {d}t} {Z}(t)\Biggr ) {X_p}(t) \\&\quad + {Z}(t) \Biggl [ {DF}_1(t){X_p}(t) + {DF}_2(t) \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) {P}(t-s) \Biggr ] \\&\quad + \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s)~ {Z}(t+s) ~ {DF}_2(t+s) ~ {X_p}(t) \\&\quad - ~ {Z}(t) ~ {DF}_2(t) ~ \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) ~ {X_p}(t-s)=~ 0\\ \end{aligned}$$

We factorize the bracket and obtain:

$$\begin{aligned}&\Leftrightarrow \Biggl (\frac{\mathrm {d}}{\mathrm {d}t} {Z}(t)\Biggr ) {X_p}(t) \\&\quad + {Z}(t){F}_1(t){X_p}(t) \\&\quad + {Z}(t){F}_2(t) \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) {P}(t-s)\\&\quad + \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s)~ {Z}(t+s) ~ {F}_2(t+s) ~ {X_p}(t) \\&\quad - ~ {Z}(t) ~ {F}_2(t) ~ \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s) ~ {P}(t-s)=~ 0\\ \end{aligned}$$

The third and the fifth term cancel, and we obtain:

$$\begin{aligned} \Leftrightarrow \Biggl [\underbrace{\frac{\mathrm {d}}{\mathrm {d}t} {Z}(t) + {Z}(t){DF}_1(t) + \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s)~ {Z}(t+s) ~ {DF}_2(t+s)}_{{\mathop {=}\limits ^{!}} 0}\Biggr ] {X_p}(t) =~ 0 \end{aligned}$$
(16)

Since this condition holds for arbitrary solutions \({X_p}(t)\), the bracket vanishes and we obtain the adjoint equation for distributed delay dynamics:

$$\begin{aligned}&\Rightarrow - \frac{\mathrm {d}}{\mathrm {d}t} {Z}(t) = {DF}^{T}_1(t){Z}(t) \nonumber \\&\quad + \int _{0}^{+\infty } \mathrm{d}s ~~ {\Delta }(s)~ {DF}^{T}_2(t+s) ~ {Z}(t+s) \end{aligned}$$
(17)

where superscript T denotes the transposed matrix. \(\square \)

Appendix B

1.1 Gaussian distributed delay convergence

Fig. 6
figure 6

Wilson–Cowan oscillators with Gaussian distributed delay converge to discrete delay model. The panels A, B show the phase response curves for excitatory \({Z_E}(\phi )\) and inhibitory \({Z_I}(\phi )\) populations, respectively. Panels C, D display the interaction functions for excitatory \({H_E}(\phi )\) and inhibitory \({H_I}(\phi )\) populations. Each subplot shows as the black solid curve the discrete delay solution and from bright to dark color solutions for the Gaussian distributed model for decreasing standard deviation values \(\sigma \). Parameters: \(\sigma = 0.01, 0.02, 0.03,\ldots , 0.4\) from dark to light shading \(, \tau = -1\), \(w_{ee}=20\), \(w_{ei}=21\), \(w_{ie}=16\), \(w_{ii}=6\), \(i_e=1.5\), \(i_i=-0.5\) and \(C = 0.01\)

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Fig. 7
figure 7

Wilson–Cowan oscillators with log-normal distributed delay converge to discrete delay model. The panels A, B show the phase response curves for excitatory \({Z_E}(\phi )\) and inhibitory \({Z_I}(\phi )\) populations, respectively. Panels C, D display the interaction functions for excitatory \({H_E}(\phi )\) and inhibitory \({H_I}(\phi )\) populations. Each subplot shows as the black solid curve the discrete delay solution and from bright to dark color solutions for the log-normal distributed model for decreasing standard deviation values \(\sigma \). Parameters: \(\sigma = 0.01, 0.02, 0.03,\ldots , 0.4\) from dark to light shading \(, \tau = -1\), \(w_{ee}=20\), \(w_{ei}=21\), \(w_{ie}=16\), \(w_{ii}=6\), \(i_e=1.5\), \(i_i=-0.5\) and \(C = 0.01\)

Full size image

Figure 6 represents in the panels Fig. 6A, B phase response curves for excitatory \({Z_E}(\phi )\) and inhibitory \({Z_I}(\phi )\) populations, respectively. Panels Fig. 6C, D display the interaction functions for excitatory \({H_E}(\phi )\) and inhibitory \({H_I}(\phi )\) populations. The black solid curve in each subplot shows the discrete delay solution and from bright to dark color the solutions for the Gaussian distributed model for decreasing standard deviation values \(\sigma \). This figure reveals that the Gaussian delay distribution is approaching the discrete delay solution for vanishing values of the standard deviation \(\sigma \).

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Winkler, M., Dumont, G., Schöll, E. et al. Phase response approaches to neural activity models with distributed delay. Biol Cybern 116, 191–203 (2022). https://doi.org/10.1007/s00422-021-00910-9

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