Oscillations in the bistable regime of neuronal networks

Bistability between attracting fixed points in neuronal networks has been hypothesized to underlie persistent activity observed in several cortical areas during working memory tasks. In network models this kind of bistability arises due to strong recurrent excitation, sufficient to generate a state of high activity created in a saddle-node (SN) bifurcation. On the other hand, canonical network models of excitatory and inhibitory neurons (E-I networks) robustly produce oscillatory states via a Hopf (H) bifurcation due to the E-I loop. This mechanism for generating oscillations has been invoked to explain the emergence of brain rhythms in the $\beta$ to $\gamma$ bands. Although both bistability and oscillatory activity have been intensively studied in network models, there has not been much focus on the coincidence of the two. Here we show that when oscillations emerge in E-I networks in the bistable regime, their phenomenology can be explained to a large extent by considering coincident SN and H bifurcations, known as a codimension two Takens-Bogdanov bifurcation. In particular, we find that such oscillations are not composed of a stable limit cycle, but rather are due to noise-driven oscillatory fluctuations. Furthermore, oscillations in the bistable regime can, in principle, have arbitrarily low frequency.

Figure 1

The phase diagram for the firing-rate model Eqs. (1). The transfer function for the excitatory population is given by Eq. (10), while it is linear-threshold for the inhibitory population. The solid lines are the saddle-node bifurcation lines, which meet as a cusp at $(I,J)=(1/2,1/2)$. To the left of this point there is only a single fixed point, while to the right there are three. The low-rate solution reaches zero at $I=0$ (dotted line), while the low-rate and high-rate solutions meet along the diagonal $I=1-J$ (dashed line). The feedforward inhibition $\beta =1$, while the time constant ratio $\tau =2$. There are no oscillatory instabilities, so the saddle-node lines provide a complete description of the possible bifurcations. (a)–(c) Sample bifurcation diagrams taken along the cuts shown in the phase diagram. The right panel is the frequency of the most unstable eigenvector. Stable branches are solid lines and unstable ones are dotted. Here ${\tau }_e={\tau }_i=10, J_{ii}=1, J_{ie}=J_{ei}=\sqrt{2}$.

Figure 2

The phase diagram for the same firing-rate model as in Fig. 1. Here the only change is that ${\tau }_i=100$ ($\beta =1$ as before), which is equivalent to increasing the inhibitory time constant compared to Fig. 1. Because $\tau <2\beta$, there are now two Hopf-bifurcation lines, shown in green. The saddle-node lines are the same as before. The two Hopf-bifurcation lines meet along the diagonal $I=1-J$ at a point which is approximately $(I^{*},J^{*})=(1/2+\beta ,1/2-\beta )$. This approximation is exact in the limit $\tau \to 0$. The frequency of oscillation along these lines increases to the left and reaches a maximum exactly at the point $(I^{*},J^{*})$. As $\tau \to 2\beta$, the point $(I^{*},J^{*})$ moves towards $(1/2,1/2)$ and the maximum frequency goes to zero. (a)–(e) Sample bifurcation diagrams taken along the cuts shown in the phase diagrams. Black lines are fixed points and green lines are limit- cycles. Solid lines are stable solutions and dotted lines [panel (c)] unstable ones. ${\tau }_i=100$, all other parameters are as in Fig. 1.

Figure 3

Panels (d) and (e) from Fig. 2. The close-ups show bifurcations of the low-rate fixed point.

Figure 4

An all-or-none population spike in the model Eqs. (1). The parameters are $J=1, I=0, \beta =1, \tau =0.2$, which place the model in the s-shaped regime of Fig. 3. The excitatory input is transiently increased between $t=100$ and $t=150$. Above a threshold value of the transient input an all-or-none population spike occurs. The inhibitory firing rate is shown as a dotted line.

Figure 5

An all-or-none population spike is generated here by transiently reducing the excitatory drive to the inhibitory population. The parameter values are the same as in Fig. 4

Figure 6

The phase diagram of fixed-point solutions in the vicinity of the TB bifurcation for the high firing-rate branches in the firing-rate equations [Eqs. (1)] with instantaneous synapses. Parameter values are $J_{ee}=2, J_{ei}=J_{ie}=1, J_{ii}=0, {\tau }_e={\tau }_i=20$, and $I_i=1$.

Figure 7

A bifurcation diagram in the vicinity of the Takens-Bogdanov bifurcation, showing a comparison of the amplitude equation with simulation from the rate model. Parameters are the same as in Fig. 6. The orange and black lines are from Eq. (19) and the dashed green lines are from Eq. (21).

Figure 8

(Top) The phase diagram with both synaptic delays and synaptic time constants. Here the parameter values are $\beta =1, \tau =0.2$. The excitatory and inhibitory synaptic time constants and delays are chosen so that the ratio, given by Eq. (22), remains fixed. The green curve is for the case where all synaptic time constants and delays are zero. The orange curve is for $D_e={\tau }_{r,e}={\tau }_{d,e}=0.1, D_i={\tau }_{r,i}={\tau }_{d,i}=1$ and the blue curve is for $D_e={\tau }_{r,e}={\tau }_{d,e}=1, D_i={\tau }_{r,i}={\tau }_{d,i}=10$. (Bottom) The critical frequency along the Hopf-bifurcation line for the different cases.

Figure 9

(Top) The phase diagram for the network of all-to-all connected LIF neurons. Saddle-node (SN) bifurcations occur along the black curve, whereas the colored curves are Hopf bifurcation lines. The maroon, green, and orange curves are for ${\tau }_i^r=2$, 3 and $4\mathrm{ms}$, respectively. There is no Hopf bifurcation for ${\tau }_i^r=1\mathrm{ms}$. (Bottom) The frequency of oscillation of the instability on the Hopf curves. Note how the frequency of oscillation goes to zero as the curves approach the SN, indicating the vicinity of a Takens-Bogdanov bifurcation. Other parameters are ${\tau }_e^r=1\mathrm{ms}, {\tau }_e^d={\tau }_i^d=10\mathrm{ms}, {\tau }_e=20\mathrm{ms}, {\tau }_i=10\mathrm{ms}, {\tau }_r=1\mathrm{ms}, J_{ei}=J_{ie}=30\mathrm{mV}$/ms, $J_{ii}=20\mathrm{mV}$/ms, $I_{ext}^I=-60\mathrm{mV}$/ms, ${\sigma }_e={\sigma }_i=5\mathrm{mV}, V_t=-55\mathrm{mV}$, and $V_r=-65\mathrm{mV}$.

Figure 10

A magnification of the phase diagram from Fig. 9 and illustrative bifurcation diagrams and corresponding dynamics from numerical simulations (b)–(d). The phase diagram shows the line of SN bifurcations (black) and a line of H bifurcations for ${\tau }_i^r=2\mathrm{ms}$ (maroon). The vertical dashed lines indicate the values of the recurrent excitatory synaptic weights $J_{ee}=37$, 38, and 39 mV/ms and the ranges of external input ${\mu }_{e,ext}\in [-54,-59]\mathrm{mV}$ used for the bifurcation diagrams and numerical simulations shown in (b)–(d). (b) (Top) The bifurcation diagram for $J_{ee}=37\mathrm{mV}$/ms showing the stationary firing rate ${\nu }_{e0}$ of the excitatory population and the frequency $\omega$ of the most unstable eigenvalue, both calculated from the dispersion equation for the network of LIF neurons. Below this diagram is the firing rate of the E neurons from a numerical simulation in which the external input is ramped down across the range of values from the bifurcation diagram. (c) and (d) Same as (b), but for $J_{ee}=38$ and $39\mathrm{mV}$/ms. All parameter values are the same as in Fig. 9.

Figure 11

Oscillatory fluctuations in the bistable regime in the $\delta$ and $\theta$ ranges. (a) A bifurcation diagram given the same parameters as in Fig. 9 and with $J_{ee}=39\mathrm{mV}$/ms for the case when there is no Hopf bifurcation, i.e., oscillatory modes are damped. (b) Examples of sustained activity in the bistable regime for ${\mu }_{e,ext}=-57.78\mathrm{mV}$ [see vertical dotted line in panel (a)]. The black trace is for slow excitation: ${\tau }_e^r={\tau }_i^r=1\mathrm{ms}, {\tau }_e^d=20\mathrm{ms}, {\tau }_i^d=10\mathrm{ms}$. In this case oscillations are very strongly damped. The orange trace is for the same parameters except ${\tau }_e^d=10\mathrm{ms}$. (c) The power spectra for 10-s traces of sustained activity for the traces shown above.

Figure 12

Oscillatory fluctuations in the bistable regime in the low-$\beta$ range. (a) The firing rate of the excitatory population in response to a transient input at $t=1000\mathrm{ms}$ and for three values of ${\tau }_e^d$: 10 ms (black), 4ms (orange), and 2 ms (violet). Note that when ${\tau }_e^d=2\mathrm{ms}$ the upper branch is unstable and the response is a population burst. Parameters: $I_{i,ext}=-48.3\mathrm{mV}, J_{ee}=10\mathrm{mV}$/ms, $J_{ie}=20\mathrm{mV}$/ms, $J_{ei}=J_{ii}=3\mathrm{mV}, {\tau }_e^r={\tau }_i^r={\tau }_i^d=1\mathrm{ms}, {\tau }_e=20\mathrm{ms}, {\tau }_i=10\mathrm{ms}, {\tau }_r=1\mathrm{ms}, V_t=-55\mathrm{mV}, V_r=-59\mathrm{mV}$.

Figure 13

The response of the network of LIF neurons to transient inputs in the bistable regime with unstable oscillations. (a) The bifurcation diagram for ${\tau }_e^r=1\mathrm{ms}, {\tau }_i^r=2\mathrm{ms}, {\tau }_e^d={\tau }_i^d=10\mathrm{ms}$. All other parameters are the same as in Fig. 9. The vertical dashed line shows the value of the input used for the simulations in (b). (b) Sample traces of the excitatory firing rate for different amplitudes of a transient input given from 1000 to $1050\mathrm{ms}$. The color code from smallest to largest amplitude: black, red, violet, orange, green, black (dashed).