Bursting Reverberation as a Multiscale Neuronal Network Process Driven by Synaptic Depression-Facilitation

Neuronal networks can generate complex patterns of activity that depend on membrane properties of individual neurons as well as on functional synapses. To decipher the impact of synaptic properties and connectivity on neuronal network behavior, we investigate the responses of neuronal ensembles from small (5–30 cells in a restricted sphere) and large (acute hippocampal slice) networks to single electrical stimulation: in both cases, a single stimulus generated a synchronous long-lasting bursting activity. While an initial spike triggered a reverberating network activity that lasted 2–5 seconds for small networks, we found here that it lasted only up to 300 milliseconds in slices. To explain this phenomena present at different scales, we generalize the depression-facilitation model and extracted the network time constants. The model predicts that the reverberation time has a bell shaped relation with the synaptic density, revealing that the bursting time cannot exceed a maximum value. Furthermore, before reaching its maximum, the reverberation time increases sub-linearly with the synaptic density of the network. We conclude that synaptic dynamics and connectivity shape the mean burst duration, a property present at various scales of the networks. Thus bursting reverberation is a property of sufficiently connected neural networks, and can be generated by collective depression and facilitation of underlying functional synapses.


Introduction
In this supplementary material, we first present the tables of parameters used in the model for neuronal island and hippocampal slices. We then show that the level of the noise is not enough to generate spontaneous bursting at a time scale of minutes. Finally, we show that blocking the metabolism of astrocyte does not affect bursting reverberation. Finally, we show the analytical computation of the reverberation time as a function of the synaptic connectivity J.

Figure A. Reverberation bursting ratio when the interval between pulses varies.
Using the parameters for culture (see table 1, the ratio converges to one after ten seconds.

Derivation of formula [5]: Analytical estimation of the reverberation time T R
We present in this section our analytical computation of the reverberation time as a function of the synaptic connectivity J (formula 5 of the main manuscript). This reverberation time T R is defined using the firing rate variable h, as the duration of the bursting activity above a certain threshold h th , induced here by a single spike. During the reverberation period, the firing rate remains approximatively constant in the initial phase of the response, which allows us to partially decouple the synaptic equations (system 1 in the main text), that we recall now Our goal is to estimate T R as a function of the threshold h th .

Approximation procedure
During the early bursting time, we approximate equation 2 and 3 of system (1), by considering h(t) ≈ H, its initial value and obtain the new approximated system Indeed, the firing rate h depends on the facilitation and depression variables x and y respectively. Although this approximation can affect drastically their dynamics, later on in the decay phase, it will not change the return to equilibrium of the firing rate h. In

Analysis of the approximated system
Because in system (2) the dynamics of x and y do not depend anymore on h, we can now integrate them and obtain where and Because the function f increases with the time, we further approximate leading with equation 3 to This approximation is quite robust as demonstrate by figure E. Finally, using expressions 3 and 8, we obtain for the firing rate h . (9)

Approximation of the firing rate h
To obtain an explicit expression for the firing rate variable h, we now decompose the last term into two parts

+ t d LHx(s) ds .(10)
The first term I is Because the term I is the sum of two integrals of decreasing functions, we use Laplace's method at the point 0, which is a regular. Thus using relations Furthermore, Finally, We shall now estimate II using that t d LH = 0.54 < 1 and x(t) < 1. Expanding in Taylor series yields where we have used equation 3. Finally, we obtain that where Summarizing the previous estimates, a power series expansion in the variable where C = −t d LHA. Reorganizing the series by changing the order of summation, we get Using the value of the parameters, the variable B and B A are small and Thus, we shall neglect terms of order greater than 2 in order of B and B A to obtain Finally using equation 9, we obtain an approximated expression for the rate h

The reverberation time satisfies a transcendental equation
We derive here a transcendental for the reverberation time T R . For that purpose, we follow the experimental protocol where an induced spike (at time zero) sets the firing rate to a value H. The reverberation time T R is then defined as the first time where the firing rate reaches the threshold h th that is We can now use expression 22 to estimate the reverberation T R as a function of the synaptic and network parameters: At this stage, we conclude that the reverberation time T R is solution of a transcendental equation ] . (25)

Analytical approximation of the reverberation time
To obtain an explicit expression for the reverberation time T R as function of J, we shall expand the exponential terms in the transcendental equation, which can be written as where When the reverberation time T R is short enough, we can Taylor expand the function G to second order polynomial, denoted by P (t), where Using A + B = X and αB = −XKH, we obtain that P (t) = Xt + 1 2 With a = t f KH and b = t d LHX, we have B = − Xa 1 + a and C = −b .
Thus, we obtain that