Slow oscillations in neural networks with facilitating synapses

Abstract

The synchronous oscillatory activity characterizing many neurons in a network is often considered to be a mechanism for representing, binding, conveying, and organizing information. A number of models have been proposed to explain high-frequency oscillations, but the mechanisms that underlie slow oscillations are still unclear. Here, we show by means of analytical solutions and simulations that facilitating excitatory (E _f) synapses onto interneurons in a neural network play a fundamental role, not only in shaping the frequency of slow oscillations, but also in determining the form of the up and down states observed in electrophysiological measurements. Short time constants and strong E _f synapse-connectivity were found to induce rapid alternations between up and down states, whereas long time constants and weak E _f synapse connectivity prolonged the time between up states and increased the up state duration. These results suggest a novel role for facilitating excitatory synapses onto interneurons in controlling the form and frequency of slow oscillations in neuronal circuits.

Appendices

Appendix A—Stability of dynamics with a “frozen” J _ie

When short time scales are considered, the stability of Eq. (1) can be determined as a function of the parameter J _ie . We use Eq. (1) in a matrix form for the supra-threshold linear regime.

$$ \begin{aligned} & {\left( \begin{aligned} & {\mathop E\limits^. } \\ & {\mathop I\limits^. } \\ \end{aligned} \right)} = A{\left( \begin{aligned} & E \\ & I \\ \end{aligned} \right)} + {\left( \begin{aligned} & \tfrac{\beta } {{\tau _{e} }}{\left( {E_{0} - T} \right)} \\ & \tfrac{\beta } {{\tau _{i} }}{\left( {I_{0} - T} \right)} \\ \end{aligned} \right)} \\ & A = {\left( {\begin{array}{*{20}c} {{\tfrac{{\beta J_{{ee}} - 1}} {{\tau _{e} }}}} & {{\tfrac{{ - \beta J_{{ei}} }} {{\tau _{e} }}}} \\ {{\tfrac{{\beta J_{{ie}} }} {{\tau _{i} }}}} & {{\tfrac{{ - \beta J_{{ii}} - 1}} {{\tau _{i} }}}} \\ \end{array} } \right)} \\ \end{aligned} $$

(A.1)

The condition for stability is a negative trace and positive determinant. Since Trace(A) is independent of J _ie , we chose the simulation parameters (see Table 1) which render it negative, thus allowing stability for some J _ie values. The stability is thus determined by Det(A), defining $J_{ie}^{th}$ as a border between the stable and the unstable regimes:

$$J_{ie}^{th} = \frac{{\left( {\beta J_{ee} - 1} \right)\left( {\beta J_{ii} + 1} \right)}}{{\beta ^2 J_{ei} }}$$

(A.2)

$J_{ie}^{th}$ represents the criterion for stability as follows: when J _ie >$J_{ie}^{th}$, E and I relax to a constant, and when J _ie >$J_{ie}^{th}$, the solution diverges.

Appendix B—estimating the oscillation borders

The right border of the oscillation regime (J _R in Fig. 2(a)) can be calculated by searching for the $J_{ie}^0$ value for which the steady state becomes stable. This can be done by calculating the eigenvalues of the linearized dynamics at the steady state:

$$\left( {\begin{array}{*{20}c} {\frac{{\beta J_{ee} - 1}}{{\tau _e }}}&{\frac{{ - \beta J_{ei} }}{{\tau _e }}}&0&0 \\ {\frac{{\beta J_{ie}^0 x_{ss} u_{ss} }}{{\tau _i }}}&{\frac{{ - \left( {\beta J_{ii} + 1} \right)}}{{\tau _i }}}&{\frac{{\beta J_{ie}^0 x_{ss} E_{ss} }}{{\tau _i }}}&{\frac{{\beta J_{ie}^0 u_{ss} E_{ss} }}{{\tau _i }}} \\ {U\left( {1 - u_{ss} } \right)}&0&{ - \left( {\frac{1}{{\tau _f }} + E_{ss} U} \right)}&0 \\ { - x_{ss} u_{ss} }&0&{ - x_{ss} E_{ss} }&{ - \left( {\frac{1}{{\tau _r }} + E_{ss} u_{ss} } \right)} \\ \end{array} } \right),$$

(B.1)

where x _ss , u _ss , E _ss are the steady state values.

The left border of the oscillation regime (J_L in Fig. 2(a)) is defined by a divergence of the oscillations, and was thus searched for by simulating the network until the border between a stable limit cycle and divergent behavior was found.