Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities

Abstract

We investigate how synchrony can be generated or induced in networks of electrically coupled integrate-and-fire neurons subject to noisy and heterogeneous inputs. Using analytical tools, we find that in a network under constant external inputs, synchrony can appear via a Hopf bifurcation from the asynchronous state to an oscillatory state. In a homogeneous net work, in the oscillatory state all neurons fire in synchrony, while in a heterogeneous network synchrony is looser, many neurons skipping cycles of the oscillation. If the transmission of action potentials via the electrical synapses is effectively excitatory, the Hopf bifurcation is supercritical, while effectively inhibitory transmission due to pronounced hyperpolarization leads to a subcritical bifurcation. In the latter case, the network exhibits bistability between an asynchronous state and an oscillatory state where all the neurons fire in synchrony. Finally we show that for time-varying external inputs, electrical coupling enhances the synchronization in an asynchronous network via a resonance at the firing-rate frequency.

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Notes

  1. 1.

    Clearly, the amount of charge transferred during any portion of the spike is proportional to γ gap . Here we deliberately treat β gap and γ gap as independent parameters to be able to consider the case γ gap  = 0 while β gap  ≠ 0.

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Acknowledgements

This work was supported by the Agence Nationale de la Recherche grant ANR-05-NEUR-030.

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Correspondence to Srdjan Ostojic.

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Action Editor: Alain Destexhe

Appendix

Appendix

A Networks of exponential integrate-and-fire neurons

In order to study the dynamics analytically, we used a model based on leaky integrate-and-fire (LIF) neurons. An important caveat of this approach is that the voltage trace during the action potential is not modeled. To account for electrotonic coupling during the action potential, we assumed that the transmission of the depolarizing part is instantaneous. While this approximation is supported by experimental measurements in fast-spiking cells (Galarreta and Hestrin 2001a), it is important to show that the obtained results, and in particular the bistability between synchrony and asynchrony, are not artifacts of this approximation. In this section, we consider the exponential integrate-and-fire model, which includes a spike-generating sodium current, and thus produces full traces of action potentials. We briefly consider the spikelets produced in this model by the transmission of the action potentials through electrical synapses, and then show that bistability between synchrony and asynchrony is still found in this model.

The exponential integrate-and-fire model is a non-linear integrate-and-fire model that has been shown to approximate remarkably well the action potential shape, and the dynamic properties of more complex conductance-based models such as the Hodgkin-Huxley model (Fourcaud-Trocmé et al. 2003). The dy namics of the membrane potential of neuron i are given by

$$\begin{array}{lll} \tau \frac{\partial V_i}{\partial t}&=& -V_i+\Delta_T\, \mathrm{exp}\left(\frac{V_i-V_T}{\Delta_T}\right)+\frac{g_c}{N}\sum\limits_{j\neq i}V_j\\ &&+\,\mu_{ext}+ \sigma \sqrt{\tau}\eta_i(t) \end{array} $$
(36)

This model includes an exponential spike-generating current: once the membrane potential crosses the threshold V T , it diverges to infinity in finite time. This divergence represents the firing of an action potential. After the divergence, in the original EIF model, the membrane potential is reset instantaneously to V r . Since this gives an unrealistic spike shape, we reset the membrane potential to V r following a linear trajectory during a refractory period τ rp , as was done in Dugué et al. (2008).

One advantage of the EIF model over more complicated models is that it provides a way of modulating the shape of the generated action potential by modifying a single parameter Δ T . The parameter Δ T determines the sharpness of spike-initiation: the larger Δ T , the slower the spike initiation, and the wider the full spike trace. The width of the hyperpolarizing part of the spike can be controlled via the refractory period τ rp . In Fig. 19 we illustrate the effect on a postsynaptic cell of spikes of different width: a wide spike leads to a predominantly excitatory spikelet, while a narrow one elicits a mostly inhibitory spikelet. Note that the shapes of the spikelets are very similar to the ones obtained in the simplified, leaky integrate-and-fire model (cf. Fig. 1).

Fig. 19
figure19

Spikelets elicited in a post-synaptic cell by the transmission through an electrical synapse of a pre-synaptic spike, obtained from the exponential integrate-and-fire model for two different sets of coupling parameters. (A) A wide presynaptic spike leads to a dominantly excitatory spikelet (V T  = 5.1 mV, Δ T  = 4 mV, τ rp  =  3 ms, V r  = − 10 mV, holding potential 3 mV); (B) a narrow presynaptic spike leads to a dominantly inhibitory spikelet (Δ T  = 1 mV, τ rp  =  1 ms, V r  = − 10 mV, holding potential 0 mV)

In Fig. 20, we illustrate the fact that bistability between synchrony and asynchrony is generically found in networks of exponential integrate-and-fire neurons. We display the activity of a network consisting of N = 50 neurons. In such a small system, the activity switches spontaneously between the asynchronous and the synchronous state. In this particular example, the firing of neurons in the synchronous state is not as highly coordinated as in the case of LIF neurons displayed in Fig. 9, but the two states can be clearly distinguished.

Fig. 20
figure20

Bistability in a network of exponential integrate-and-fire neurons: spontaneous transitions between synchrony and asynchrony in a network of N = 50 neurons. (A) Spike raster of the full network; (B) instantaneous population firing rate ν(t) computed in 1 ms bins. The parameters used in the simulation are: μ ext  = 3 mV, σ = 0.47 mV, g c  = 0.5, V T  = 5.1 mV, Δ T  = 1 mV, V r  = − 10 mV, τ rp  = 1.7 ms

B Linear stability of the asynchronous state in the homogeneous case

The probability distribution P(V,t) obeys

$$\begin{array}{lll} \tau \frac{\partial P(V,t)}{\partial t}&=&\frac{\partial}{\partial V}\left[(V-\mu_{syn}-\mu_{ext})P(V,t)\right]\\ &&+\frac{\sigma^2}{2}\frac{\partial^2}{\partial V^2} P(V,t), \end{array} $$
(37)

where \(\mu_{syn}=g_c\langle V \rangle+\beta \tau \nu(t)\). The normalization and boundary conditions are:

$$ \int_{-\infty}^{V_{th}} P(V,t) dV=1 $$
(38)
$$ \int_{-\infty}^{V_{th}} V P(V,t) dV=\langle V(t) \rangle $$
(39)
$$ \frac{\partial P}{\partial V}(V_{th},t)=-\frac{2 \nu(t) \tau}{\sigma^2} $$
(40)
$$ \frac{\partial P}{\partial V}(V_r^+,t)-\frac{\partial P}{\partial V}(V_r^-,t)=-\frac{2 \nu(t) \tau}{\sigma^2}. $$
(41)

Multiplying Eq. (37) by V and integrating, we obtain an equation for the evolution of the mean membrane potential \(\langle V \rangle\):

$$ \tau \frac{d \langle V\rangle}{dt}=(g_c-1)\langle V\rangle + \mu_{ext} + (\beta-(V_{th}-V_r))\tau \nu(t). $$
(42)

From this point on, the analysis follows the same steps as in Brunel and Hakim (1999). We use the rescaled variables v(t), n(t) and m syn (t) defined by:

$$\begin{array}{lll} \langle V(t) \rangle&=& V_0(1+v(t)),\qquad \nu(t)=\nu_0(1+n(t)),\\ \mu_{syn}(t)&=&\mu_{syn,0}+m_{syn}(t), \end{array} $$
(43)
$$ \mu_{syn,0}=g_c V_0+\beta \tau \nu_0,\qquad P=\frac{2\tau \nu_0}{\sigma}Q, $$
(44)
$$\begin{array}{lll} y{\kern4pt}&=&\frac{V-\mu_{syn,0}-\mu_{ext}}{\sigma}, \qquad y_{th}=\frac{V_{th}-\mu_{syn,0}-\mu_{ext}}{\sigma},\\ y_r&=&\frac{V_r-\mu_{syn,0}-\mu_{ext}}{\sigma}. \end{array} $$
(45)

With these notations, Eq. (37) becomes

$$ \tau \partial_t Q(y,t)=\mathcal{L}[Q]- \frac{m_{syn}}{\sigma} \frac{\partial}{\partial y} Q, $$
(46)

where the linear operator \(\mathcal{L}\) is defined as

$$ \mathcal{L}[Q]=\frac{1}{2}\frac{\partial^2}{\partial y^2} Q+\frac{\partial}{\partial y} (yQ) $$
(47)

and the boundary conditions read

$$ Q(y_{th},t)=0, \qquad \frac{\partial Q}{\partial y}(y_{th},t)=-(1+n(t)) $$
(48)
$$ \left[Q\right]^{y_{r}^{+}}_{y_{r}^{-}}=0, \qquad \left[\frac{\partial Q}{\partial y}\right]^{y_{r}^{+}}_{y_{r}^{-}}=-(1+n(t)). $$
(49)

B.1 Stationary state

The steady-state solution of Eq. (46) is given by

$$ Q_0(y)=\left\{ \begin{array}{ll} \exp(-y^2)\int_y^{y_{th}} du\, \exp(u^2) & y>y_r\\[4pt] \exp(-y^2)\int_{y_r}^{y_{th}} du\, \exp(u^2) & y<y_r. \end{array}\right. $$
(50)

The stationary mean membrane potential V 0 is given by Eq. (42):

$$ V_0=\frac{\mu_{ext}+\tau \nu_0(\beta-(V_{th}-V_r))}{1-g_c}. $$
(51)

The stationary firing rate ν 0 is then obtained from the normalization condition Eq. (38):

$$ \int_{y_r}^{y_{th}} du\,e^{u^2} \int_{-\infty}^{u}dv e^{-v^2}=\frac{1}{2\tau \nu_0}. $$
(52)

As y r and y th depend on ν 0, Eq. (52) is an implicit equation which can be solved self-consistently for ν 0.

B.2 Linear stability

In the following, we expand the time varying quantities perturbatively at successive orders:

$$\begin{array}{rll} Q(y,t)&=&Q_0(y)+Q_1(y,t)+Q_2(y,t)+\ldots \\ n(t)&=&n_1(t)+n_2(t)+\ldots\\ v(t)&=&v_1(t)+v_2(t)+\ldots\\ m_{syn}(t)&=&m_{syn,1}(t)+m_{syn,2}(t)+\ldots \end{array} $$
(53)

At first order Eq. (46) becomes

$$ \tau \partial_t Q_1(y,t)=\mathcal{L}[Q_1]-\frac{m_{syn,1}}{\sigma} \frac{\partial}{\partial y} Q_0. $$
(54)

The eigenmodes of Eq. (54) are of the form

$$\begin{array}{rll} Q_1(y,t){\kern4pt} &=&\exp\left(\frac{\lambda t}{\tau}\right)\tilde n_1(\lambda) \hat Q_1(y,\lambda)\\ n_1(t)&=&\exp\left(\frac{\lambda t}{\tau}\right)\tilde n_1(\lambda)\\ v_1(t)&=&\exp\left(\frac{\lambda t}{\tau}\right) \tilde v_1(\lambda)\\ m_{syn,1}(t)&=&\exp\left(\frac{\lambda t}{\tau}\right)\tilde m_1(\lambda) \end{array}$$

As v(t) and n(t) are linearly related via Eq. (42)

$$ \tilde v_1 (\lambda)=\frac{\tau \nu_0}{V_0} \frac{\beta-(V_{th}-V_r)}{1+\lambda-g_c}\tilde n_1 (\lambda) $$
(55)

and

$$ \tilde m_{1} (\lambda)=\tau \nu_0 R_g(\lambda) \tilde n_1 (\lambda) $$
(56)

with

$$ R_g(\lambda)=\frac{\beta(1+\lambda)-g_c(V_{th}-V_r)}{1+\lambda -g_c}. $$
(57)

\(\hat Q_1\) therefore obeys the ordinary differential equation

$$ \lambda \hat Q_1(y,\lambda)=\mathcal{L}[\hat Q_1](y,\lambda)-\frac{\tau \nu_0 R_g(\lambda)}{\sigma} \frac{\partial}{\partial y} Q_0, $$
(58)

together with the boundary conditions

$$ \hat Q_1(y_{th},\lambda)=0,\qquad \frac{\partial \hat Q_1}{\partial y}(y_{th},\lambda)=-1, $$
(59)

similar conditions at y r , and the integrability requirement

$$ \int_{-\infty}^{y_{th}} dy \hat Q_1(y,\lambda) < \infty. $$
(60)

The general solution of Eq. (58) can be written as

$$ \begin{array}{lll} && \hat Q_1(y,\lambda)\\ && {\kern1pc}=\!\left\{\! \begin{array}{ll} \alpha_1^{+}(\lambda) \phi_1(y,\lambda)\!+\!\beta_1^{+}(\lambda)\phi_2(y,\lambda)\!+\!\hat Q_1^{p}(y,\lambda) & \quad y\!>\!y_r\\ \alpha_1^{-}(\lambda) \phi_1(y,\lambda)\!+\!\beta_1^{-}(\lambda)\phi_2(y,\lambda)\!+\!\hat Q_1^{p}(y,\lambda) & \quad y\!<\!y_r \end{array}\right. \end{array} $$
(61)

where

$$ \hat Q_1^{p}(y,\lambda)=-\frac{1}{1+\lambda}\frac{\tau \nu_0 R_g(\lambda)}{\sigma} \frac{d Q_0(y)}{dy}, $$
(62)

and ϕ 1,2 are two independent solutions of the homogeneous equation

$$ \frac{1}{2}\phi''+y\phi'+(1-\lambda)\phi=0 . $$
(63)

Solutions of Eq. (63) are obtained in terms of confluent hypergeometric functions (Brunel and Hakim 1999):

$$ \phi_1(y,\lambda)=M\left[(1-\lambda)/2,1/2,-y^2\right] $$
(64)
$$\begin{array}{lll} \phi_2(y,\lambda)&=&\frac{\sqrt{\pi}}{\Gamma\left(\frac{1+\lambda}{2}\right)}M\left[(1-\lambda)/2,1/2,-y^2\right]\\ &&+\frac{\sqrt{\pi}}{\Gamma\left(\frac{\lambda}{2}\right)}2yM\left[1-\lambda/2,3/2,-y^2\right]. \end{array} $$
(65)

The boundary conditions Eq. (59) give expressions for \(\alpha_1^{+},\alpha_1^{-},\beta_1^{+}\) and \(\beta_1^{-}\) as functions of λ:

$$ \alpha_1^{+}={\tilde \phi}_2(y_{th},\lambda)\left\{1-S_2(y_{th},\lambda) \right\} $$
(66)
$$ \beta_1^{+}=-{\tilde \phi}_1(y_{th},\lambda)\left\{1-S_1(y_{th},\lambda) \right\} $$
(67)
$$ \alpha_1^{-}-\alpha_1^{+}=-{\tilde \phi}_2(y_{r},\lambda)\left\{1-S_2(y_{r},\lambda) \right\} $$
(68)
$$ \beta_1^{-}-\beta_1^{+}={\tilde \phi}_1(y_{r},\lambda)\left\{1-S_2(y_{r},\lambda) \right\}, $$
(69)

where

$${\tilde \phi}_i(y,\lambda)=\frac{\phi_i(y,\lambda)}{Wr[\phi_1,\phi_2](y,\lambda)} $$
(70)
$$ Wr[\phi_1,\phi_2](y,\lambda)=\phi_1\phi_2'-\phi_2\phi_1'=\frac{\sqrt{\pi}}{\Gamma(\lambda/2)}e^{-y^2} $$
(71)
$$ S_i(y,\lambda)=\frac{\tau \nu_0}{\sigma}\frac{R_g(\lambda)}{1+\lambda}\left(\frac{\phi_i'}{\phi_i}(y)+2y\right). $$
(72)

The eigenvalues are determined by the requirement (60) that \(\hat Q_1(y,\lambda)\) be integrable, which corresponds to \(\alpha_1^{-}=0\), i.e.

$$\begin{array}{lll} \left({\tilde \phi}_2(y_{th},\lambda)-{\tilde \phi}_2(y_{r},\lambda)\right)&=&{\tilde \phi}_2(y_{th},\lambda) S_2(y_{th},\lambda)\\ &&-{\tilde \phi}_2(y_{r},\lambda)S_2(y_{r},\lambda) \end{array} $$
(73)

or

$$ R_g(\lambda)R_n(\lambda)=1, $$
(74)

with

$$ R_g(\lambda)=\frac{(1+\lambda)\beta-g_c(V_{th}-V_r)}{(1-g_c+\lambda)} $$
(75)
$$ R_n(\lambda)=\frac{\tau \nu_0}{\sigma}\frac{1}{1+\lambda} \frac{\frac{\partial U}{\partial y}(y_{th},\lambda)-\frac{\partial U}{\partial y}(y_{r},\lambda)}{U(y_{th})-U(y_{r})} $$
(76)
$$ U(y,\lambda)=e^{y^2}\phi_2(y,\lambda). $$
(77)

B.3 Weakly non-linear analysis

Pushing the development in Eq. (53) to higher orders, it is possible to determine the non-linear contribution to the oscillations in the vicinity of the bifurcation. The n-th order terms obey inhomogeneous linear equations with forcing terms formed by quadratic combinations of lowest-order terms. The dynamics of the first-order oscillation amplitude \(\hat n_1\) is obtained from a self-consistancy condition on the third order terms, and read

$$ \tau \frac{d \hat n_1}{dt}=A \hat n_1-B |\hat n_1|^2 \hat n_1 $$
(78)

The procedure is identical to the one followed in Brunel and Hakim (1999), we therefore only give here the expressions necessary for evaluating B, and for more details refer the reader to Brunel and Hakim (1999). The results of Brunel and Hakim (1999) are recovered (in the fully connected case H = 0) by replacing below \(\bar R_g(\lambda)\) with \(-G_c e^{-\delta \lambda}\).

In the following, we write

$$ \bar R_g (\lambda)=\frac{\tau \nu_0}{\sigma} R_g(\lambda) $$
(79)
$$ \tilde W_2\left[\hat Q\right]=\frac{\hat Q \phi_2^{'}-\hat Q^{'} \phi_2}{Wr[\phi_1,\phi_2]} $$
(80)

B.3.1 Second order

The second order terms oscillate at frequencies 0 and 2ω c , and can be written as:

$$\begin{array}{lll} Q_2(y,t)&=&e^{2i\omega_c t/\tau} \hat n_1^{2}\hat Q_{2,2}(y)+e^{-2i\omega_c t/\tau} (\hat n_1^*)^{2}\hat Q_{2,2}^*(y)\\ &&+\hat Q_{2,0}|\hat n_1|^2 \end{array} $$
(81)
$$ n_2(t)=e^{2i\omega_c t/\tau} \hat n_1^{2}\rho_{2,2}+e^{-2i\omega_c t/\tau} (\hat n_1^*)^{2}\rho_{2,2}^*+|\hat n_1|^2 \rho_{2,0}. $$
(82)

\(\hat Q_{2,2}(y)\) can be expressed as

$$ \hat Q_{2,2}(y)=\left\{\! \begin{array}{ll} &{\kern-4pt} \alpha_{2,2}^{+} \phi_1(y,2i \omega_c)\!+\!\beta_{2,2}^{+}\phi_2(y,2i \omega_c)\\ &{\kern-4pt}{\kern1pc}+\!\rho_{2,2} \hat Q_{2,2}^{so}(y,i \omega_c)\!+\!\hat Q_{2,2}^{lo} \quad y\!>\!y_r\\ &{\kern-4pt} \alpha_{2,2}^{-} \phi_1(y,2i \omega_c)\!+\!\beta_{2,2}^{-}\phi_2(y,2i \omega_c)\\ &{\kern-4pt}{\kern1pc}+\!\rho_{2,2} \hat Q_{2,2}^{so}(y,i \omega_c)\!+\!\hat Q_{2,2}^{lo} \quad y\!<\!y_r, \end{array}\right. $$
(83)

where:

$$ \hat Q_{2,2}^{so}=-\frac{\bar R_g(2 i\omega_c)}{1+2i\omega_c}\frac{dQ_0}{dy} $$
(84)
$$ \hat Q_{2,2}^{lo}=\frac{\bar R_g(i\omega_c)}{1+i\omega_c}\frac{\partial Q_1}{\partial y}-\frac{\bar R_g(i\omega_c)^2}{2(1+i\omega_c)^2}\frac{d^2Q_0}{dy^2} $$
(85)
$$ \rho_{22}\!=\! \frac{\tilde W_2\left[\!\hat Q_{2,2}^{lo}\right](y_{th})\!-\!\left[\!\tilde W_2\!\left[\!\hat Q_{2,2}^{lo}\right]\!(y) \right]_{y_r^-}^{y_r^+}}{(\tilde \phi_2(y_{th})\!-\! \tilde \phi_2(y_{r}))\!-\!\tilde W_2\!\left[\!\hat Q_{2,2}^{so}\right]\!(y_{th})\!+\!\left[\!\tilde W_2\!\left[\!\hat Q_{2,2}^{so}\right]\!(y)\! \right]_{y_r^-}^{y_r^+}}. $$
(86)

Similarly, \(\hat Q_{2,0}(y)\) can be expressed as

$$ \hat Q_{2,0}(y)=\left\{ \begin{array}{ll} &\alpha_{2,0}^{+}Q_0+\beta_{2,0}^{+}\exp(-y^2)\\[4pt] &{\kern1pc}+\rho_{2,0} \hat Q_{2,0}^{so}(y){\kern25pt} +\hat Q_{2,0}^{lo} \quad y>y_r\\[4pt] &\alpha_{2,0}^{-} Q_0+\beta_{2,0}^{-}\exp(-y^2)\\[4pt] &{\kern1pc}+\rho_{2,0} \hat Q_{2,0}^{so}(y,i \omega_c)+\hat Q_{2,0}^{lo} \quad y<y_r, \end{array}\right. $$
(87)

where:

$$ \hat Q_{2,0}^{so}=-\bar R_g(0)\frac{dQ_0}{dy} $$
(88)
$$\begin{array}{lll} \hat Q_{2,0}^{lo}&=&-\left[\frac{\bar R_g(-i\omega_c)}{1-i\omega_c}\frac{\partial \hat Q_1}{\partial y}+c.c.\right]\\ &&- \frac{\bar R_g(-i\omega_c)\bar R_g(i\omega_c)}{(1+\omega_c^2)}\frac{d^2Q_0}{dy^2} \end{array} $$
(89)
$$ \rho_{20}=\frac{\left[2\frac{\bar R_g\left(-i\omega_c\right)}{1+\omega_c^2}(-\bar R_g(i\omega_c)y+1)e^{y^2}\int_{-\infty}^{y}du e^{-u^2}\right]_{y_r}^{y_{th}}} {\frac{1}{2\nu_0}- \left[\bar R_g(0)e^{y^2} \int_{-\infty}^{y}du e^{-u^2}\right]_{y_r}^{y_{th}}}. $$
(90)

B.3.2 Third order

The third order terms oscillate at frequencies ω c and 3 ω c :

$$ Q_3(y,t)=e^{3i\omega_c t /\tau} \hat Q_{3,3}(y)+e^{i\omega_c t /\tau}\hat Q_{3,1}(y) + c.c.$$
(91)
$$ n_3(t)=e^{3i\omega_c t /\tau}\hat n_{3,3}(y)+e^{i\omega_c t /\tau}\hat n_{3,1}(y) + c.c. $$
(92)

The part \(\hat Q_{3,1}(y)\) of Q 3(y,t) oscillating at frequency ω c can be written as

$$ \hat Q_3(y)=\left\{ \begin{array}{ll} &{\kern-4pt}\alpha_3^{+} \phi_1(y,i \omega_c)+\beta_3^{+}\phi_2(y,i \omega_c)+\\[4pt] &{\kern-4pt}{\kern1pc}\hat n_{3,1} \hat Q_1^{p}(y,i \omega_c)+\hat Q_{3,1}^{lo} \quad y>y_r\\[4pt] &{\kern-4pt}\alpha_3^{-} \phi_1(y,i \omega_c)+\beta_3^{-}\phi_2(y,i \omega_c)+\\[4pt] &{\kern-4pt}{\kern1pc}\hat n_{3,1} \hat Q_1^{p}(y,i \omega_c)+\hat Q_{3,1}^{lo} \quad y<y_r, \end{array}\right. $$
(93)

with

$$ \hat Q_{3,1}^{lo}=\tau \frac{d \hat n_1}{dt} \hat Q_{3,1}^d+\hat n_1 \hat Q_{3,1}^l+\hat n_1|\hat n_1|^2 \hat Q_{3,1}^c, $$
(94)

where

$$ \hat Q_{3,1}^d(y)=\left\{ \begin{array}{ll} &{\kern-4pt}\alpha_1^{+} \partial_{\lambda}\phi_1(y,i \omega_c)+\beta_1^{+} \partial_{\lambda}\phi_2(y,i \omega_c)+\\[4pt] &{\kern-4pt}{\kern1pc}\partial_{\lambda}\hat Q_1^{p}(y,i \omega_c)\ \ y>y_r\\[4pt] &{\kern-4pt}\beta_1^{-} \partial_{\lambda}\phi_2(y,i \omega_c)+\partial_{\lambda}\hat Q_1^{p}(y,i \omega_c)\\[4pt] &{\kern.8pc}y<y_r, \end{array}\right. $$
(95)
$$ \hat Q_{3,1}^l(y)=\frac{\bar R_g-\bar R_{g,c}}{1+i\omega_c}\frac{dQ_0}{dy}, $$
(96)

and

$$\begin{array}{lll} \hat Q_{3,1}^{c}\!&=&\!-\frac{\bar R_g(\!- i\omega_c)}{1\!-\!i\omega_c}\frac{\partial \hat Q_{22}}{\partial y}\!-\!\frac{\bar R_g(i\omega_c)}{1\!+\!i\omega_c}\frac{\partial \hat Q_{20}}{\partial y}\!-\!\rho_{20}\bar R_g(0)\frac{\partial \hat Q_1}{\partial y} \\ &&-\frac{\rho_{22}\bar R_g(2i\omega_c)}{1\!+\!2i\omega_c}\frac{\partial \hat Q_1^*}{\partial y}\!-\!\frac{\bar R_g(i\omega_c)^2}{2(1\!+\!i\omega_c)^2}\frac{\partial^2 \hat Q_1^*}{\partial y^2} \\ &&-\frac{\bar R_g(i\omega_c)\bar R_g(\!-i\omega_c)}{1\!+\!\omega_c^2}\frac{\partial^2 \hat Q_1}{\partial y^2}\!-\!\frac{\rho_{20}\bar R_g(0)\bar R_g(i\omega_c)}{1\!+\!i\omega_c}\frac{d^2 Q_0}{d y^2}\\ &&-\frac{\rho_{22}\bar R_g(-i\omega)\bar R_g(2i\omega_c)}{(1\!-\!i\omega_c)(2i\omega_c+1)}\frac{d^2 Q_0}{d y^2}\\ &&-\frac{\bar R_g(\!-i\omega)\bar R_g(i\omega_c)^2}{2(1\!-\!i\omega_c)(i\omega_c\!+\!1)^2}\frac{d^3 Q_0}{d y^3}. \end{array} $$
(97)

B.3.3 Final expressions

The constants A and B determining the non-linear dynamics of \(\hat n_1\) read:

$$ A = \frac{W_2[\hat Q_{3,1}^l](y_{th})-\left[\tilde W_2[\hat Q_{3,1}^l](y) \right]_{y_r^-}^{y_r^+}}{W_2[\hat Q_{3,1}^d](y_{th})-\left[\tilde W_2[\hat Q_{3,1}^d](y) \right]_{y_r^-}^{y_r^+}} $$
(98)
$$ B = \frac{W_2[\hat Q_{3,1}^c](y_{th})-\left[\tilde W_2[\hat Q_{3,1}^c](y) \right]_{y_r^-}^{y_r^+}}{W_2[\hat Q_{3,1}^d](y_{th})-\left[\tilde W_2[\hat Q_{3,1}^d](y) \right]_{y_r^-}^{y_r^+}}, $$
(99)

Note that the denominator in these expressions can be rewritten as

$$\begin{array}{lll} &&{\kern-6pt}\tilde W_2[\hat Q_{31}^d](y_{th})-\left[\tilde W_2[\hat Q_{31}^d](y) \right]_{y_r^-}^{y_r^+}\\ &&{\kern6pt}=-(\tilde \phi_2(y_{th})- \tilde \phi_2(y_{r}))\frac{\partial}{\partial \lambda}\bar R_gR_n (i\omega_c). \end{array} $$
(100)

C The heterogeneous case

We consider a distribution of input currents such that neurons in the network receive a mean external current \(\mu_{ext}=\bar \mu_{ext} + z\, \delta \mu\) with probability η(z). Taking into account the interactions, the total current received by a neuron becomes

$$ \mu_{tot}(z)={\bar \mu}_{ext} + z\, \delta \mu + \beta \tau \bar \nu (t)+g_c \bar{\langle V \rangle} (t) $$
(101)

where \({\bar \nu}\) and \(\bar{\langle V \rangle}\) are the mean firing rate and membrane potential in the network, averaged over the heterogeneity. We denote by \(\langle \, \rangle\) the average over neurons with the same z, and by a bar the average over different z.

C.1 Stationary state

In the stationary state, the firing rate of the neuron with total input \(\mu_{tot, 0}(z)={\bar \mu}_{ext} + z\, \delta \mu + \beta \tau \bar \nu_0 +g_c \bar V_0\) is given by

$$ \nu_0(z)=\left(2 \tau \int_{\frac{V_{th}-\mu_{tot, 0}(z)}{\sigma}}^{\frac{V_{r}-\mu_{tot, 0}(z)}{\sigma}} du\,\,\, e^{u^2} \int_{-\infty}^{u} dv\,\,\, e^{-v^2}\right)^{-1}, $$
(102)

where μ tot, 0(z) depends on the stationary average firing rate \(\bar \nu_0\) and mean membrane potential \(\bar V_0\), which are unknown at this stage.

Averaging over heterogeneity, we get

$${\bar \nu_0}=\int dz\, \eta(z) \nu_0(z) $$
(103)
$$\begin{array}{lll} {\bar V_0}&=&\int dz \,\,\, \eta(z) V_0(z)\,\\ &=&\frac{1}{1-g_c}\left({\bar \mu}_0+\beta \tau {\bar \nu_0}+\tau {\bar \nu_0} (V_r-V_{th})\right). \end{array} $$
(104)

These equations can be solved self-consistently for \(\bar \nu_0\) and \(\bar V_0\). The distribution ρ of firing rates in the network is then obtained as

$$ \rho(\nu)=\int dz \,\eta(z)\,\delta(\nu-\nu_0(z)) $$
(105)

C.2 Linear stability

To determine the linear stability of the asynchronous state, we follow the same steps as in Appendix A. We start by linearizing all the time-dependent quantities around their stationary value:

$$ Q(y,z,t)=Q_0(y,z)+Q_1(y,z,t) $$
(106)
$$ \bar{\langle V \rangle}=V_0(1+{\bar v}(t)) $$
(107)
$$ \nu(z)=\nu_0(z)(1+n(z,t)) $$
(108)
$$ \bar{\nu}={\bar \nu_0}(1+{\bar n(t)}), $$
(109)

where \({\bar n(t)}=\frac{1}{{\bar \nu_0}}\int dz\,\eta(z)\nu_0(z)n(z,t)\).

At first order, after expanding Q 1, v, n and \(\bar{n}\) in eigenmodes, the Foker-Planck equation becomes

$$ \lambda \hat Q_1(y,z,\lambda)=\mathcal{L}[\hat Q_1]-\frac{\tau \bar \nu_0 R_g(\lambda) \bar n(\lambda)}{\sigma}\frac{\partial}{\partial y} Q_0 $$
(110)

with boundary conditions

$$ \hat Q_1(y_{th},z,\lambda)=0,\qquad \frac{\partial \hat Q_1}{\partial y}(y_{th},z,\lambda)=- n(z,\lambda) $$
(111)

The integrability condition for each z reads:

$$ n(z,\lambda)=R_g(\lambda)R_n(\lambda,z){\bar n(\lambda)} $$
(112)

where \(R_n(\lambda,z)= \frac{\tau \nu_0(z)}{\sigma}\frac{1}{1+\lambda} \frac{\frac{\partial U}{\partial y}(y_{th},\lambda)-\frac{\partial U}{\partial y}(y_{r},\lambda)}{U(y_{th})-U(y_{r})}\)

Multiplying both sides of Eq. (112) by η(z) ν 0(z), and integrating over z, we get

$$ R_g(i \omega){\bar R_n(i \omega)}=1, $$
(113)

with

$${\bar R_n(i \omega)}=\int d\nu \rho(\nu) R_n(i \omega, \nu). $$
(114)

D Response to external oscillations

We consider the situation where the external current varies in time as μ ext (t) = μ ext,0 + μ ext,1(t), with \(\mu_{ext,1}(t)=\tilde m_{ext}(\lambda)e^{\lambda t/\tau}+c.c.\), and λ = . In that case

$$ \tilde v_1 (\lambda)=\frac{1}{V_0} \frac{\tau \nu_0(\beta-(V_{th}-V_r))\tilde n_1(\lambda)+\tilde m_{ext}}{1+\lambda-g_c} $$
(115)

so that Eq. (58) becomes

$$ \begin{array}{lll} \lambda Q_1(y,\lambda)&=&\mathcal{L}[Q_1](y,\lambda)\\&&-\frac{\tau \nu_0 R_g(\lambda)+\frac{1+\lambda}{1+\lambda-g_c}\frac{\tilde m_{ext}(\lambda)}{\tilde n(\lambda)}}{\sigma} \frac{\partial}{\partial y} Q_0. \end{array} $$
(116)

The condition \(\alpha_1^{-}\) then gives

$$ \left(R_g+ \frac{1+\lambda}{1+\lambda-g_c}\frac{\tilde m_{ext}(\lambda)}{\tau \nu_0 \tilde n(\lambda)} \right)R_n=1. $$
(117)

Rearranging the terms, we obtain

$$ \tau \nu_0 \tilde n(\lambda)=\frac{1+\lambda}{1+\lambda-g_c}\frac{R_n(\lambda)}{1-R_n(\lambda) R_g(\lambda)}\tilde m_{ext}(\lambda) $$
(118)

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Ostojic, S., Brunel, N. & Hakim, V. Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J Comput Neurosci 26, 369 (2009). https://doi.org/10.1007/s10827-008-0117-3

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Keywords

  • Gap junctions
  • Oscillations
  • Neural networks