Phase-locking and bistability in neuronal networks with synaptic depression

Abstract

We consider a recurrent network of two oscillatory neurons that are coupled with inhibitory synapses. We use the phase response curves of the neurons and the properties of short-term synaptic depression to define Poincaré maps for the activity of the network. The fixed points of these maps correspond to phase-locked modes of the network. Using these maps, we analyze the conditions that allow short-term synaptic depression to lead to the existence of bistable phase-locked, periodic solutions. We show that bistability arises when either the phase response curve of the neuron or the short-term depression profile changes steeply enough. The results apply to any Type I oscillator and we illustrate our findings using the Quadratic Integrate-and-Fire and Morris-Lecar neuron models.

Keywords: Bistability; Coupled Oscillators; Phase Response Curve; Short-Term Synaptic Depression; Two-dimensional Poincarè Map.

Figure 1:

PRC due to synaptic input. A. The PRCs obtained from the QIF model (1) for different synaptic strengths. B. The PRCs obtained from the ML model (2) for different synaptic strengths.

Figure 2:

Steady state synaptic plasticity profiles r_ss(P ) for the two synaptic models. A. The plasticity profile for the QIF model (6). B. The plasticity profile for the ML model (11). Notice the difference in the scaling of the y-axes; the value of the depression variable changes over a much larger range for the synapse considered for the ML model.

Figure 3:

The variables that are used to define the Poincaré maps are shown on the simulation of ML neurons. The cycle length P_n of cell A in cycle n (measured when voltage crosses V_th) can be divided into the delay between cell A activity to cell B activity (dt_n) and the opposite (dτ_n). The cycle period Q_n of cell B in cycle n is dτ_n+dτ_n+1.

Figure 4:

Phase locking of QIF neurons with static synapses. A. Steady-state intrinsic phase of neuron A obtained from the map Π given in equation (22) as a function of the synaptic coupling strength ${\overline{g}}_{B\to A}$. B. The relationship $g_{B\to A}=h\left({\overline{g}}_{B\to A}\right)$ between the synaptic strengths of the static map Π and the depressing map Π^dyn given in equation (26) obtained from equations (33–35). The dashed horizontal line at g_B→A = 5.35 intersects $h\left({\overline{g}}_{B\to A}\right)$ at three points (inside circles). Two of these are points that correspond to bistability in the presence of synaptic depression.

Figure 5:

Fixed points of the 2-D map Π^dyn given in Equation (26) and their equivalence with the fixed points of the map Π given in Equation (22). A. Steady-state intrinsic phase of neuron A obtained from Π^dyn as a function of $g_{B\to A}$. Dashed vertical line at g_B→A = 5.35 lies within the region of bistablity which is also shown in the inset. B. The steady-state value of the depression variable obtained from Π^dyn as a function of $g_{B\to A}$. C. The steady-state intrinsic phase of neuron A obtained from the depressing map Π^dyn is equivalent to the phase obtained from the static map Π (compare with Figure 4A) when plotted as a function of ${\overline{g}}_{B\to A}{=}_{g_{B\to A}}{r}^{*}$ = g_B→Ar*. D. The steady state activity phase ${\tilde{\varphi }}^{*}$ of neuron A obtained from Π^dyn.

Figure 6:

The dependence of the eigenvalues µ₁ and µ₂ of the depressing map (26) on the parameter g_B→A. The absolute values of µ₁ and µ₂ are shown in orange and purple, respectively. These values overlap for complex eigenvalues (blue box). The fixed points are stable when both eigenvalues are less than 1 in absolute value. The fixed points lose (regain) stability via saddle-node bifurcation when µ₁ is greater (less) than 1 (green circles).

Figure 7:

The dependence of bistability in QIF neurons on the parameters that govern synaptic depression (5). The unstable region increases and the region of bistability changes as f is decreased from left to right. A similar change is observed as τ_r is increased.

Figure 8:

Existence of bistability in ML neurons depending on the steady state plasticity profile. A. The steady-state intrinsic phase of neuron A obtained from the static map Π given in equation (22) as a function of the synaptic conductance ${\overline{g}}_{B\to A}$. B. Different steady state plasticity profiles r_ss used in the depressing map Π^ss given in equation (31). C. The relationship between the synaptic conductances of the static map Π and the depressing map Π^ss for different r_ss. D. The steady-state intrinsic phase ϕ* of neuron A obtained from the map Π^ss for different r_ss as a function of the synaptic conductance g_B→A. Notice that the bistability region exists only when a sharp steady state profile is used.

Figure 9:

Simulation of coupled ML neurons compared with fixed points of the map Π^ss (31). A. Membrane voltages of two ML neurons coupled with inhibitory synapses when the B to A synapse is depressing. The network locks at two different phases. From t = 900 to 1, 100, neuron B is hyperpolarized, causing the network to switch to the other phase-locked solution. B. Activity phase versus synaptic conductance obtained from the map Π^ss. C. The evolution of the synaptic variables from neuron B to A. D. Period of neuron A (also neuron B) versus synaptic conductance obtained from the map Π^ss. The Z-shaped (S-shaped) curve in panel B (panel D) shows the different phases (periods) that exist over a range of conductance values of g_B→A. The lower and upper branches represent stable solutions, while the dotted middle branch represents unstable solutions. The simulations in the left two panels occur for g_B→A = 0.125, where for t < 900, the phase-locked solution corresponds to a point on the upper (lower) branch, and for t > 1100, the solution converges to a phase-locked solution on the lower (upper) branch.

Figure 10:

Fixed points of the map Π^ss (31). A–C. The creation of fixed points of the map for different sets of parameters. The surfaces ${\Pi }_1^{ss}\left(\varphi ,P\right)$ and ${\Pi }_2^{ss}\left(\varphi ,P\right)$ are drawn above and below the z = 0 plane denoted by the axes z₁ = ϕ and z₂ = P, respectively. The intersection of the surface $z_1={\Pi }_1^{ss}\left(\varphi ,P\right)$ with the plane z₁ = ϕ and the intersection of the surface $z_2={\Pi }_2^{ss}\left(\varphi ,P\right)$ with the plane z₂ = P yield the two black curves above and below the z = 0 plane. The fixed points of the maps lay on the intersection of the two fixed point curves whose projections C₁ and C₂ on the z = 0 plane are shown. There is one fixed point in A and C while there are three fixed points in B, depending on the value of the g_B→A. D. The projections of the fixed point curves, C₁ and C₂ are drawn on the same coordinate axes for three parameter sets. The curveC₁ is the same for all parameter sets while C₂ changes. Creation and annihilation of multiple fixed points with changing parameters is observed.