Fig 1.
Frequency response of a network of QIF neurons.
Here we show the response of an excitatory network of 104 all-to-all coupled QIF neurons with distributed input currents to periodic forcing. The model parameters are chosen such that the network is bistable, see also Fig 2A. Each panel shows the network-averaged firing rate (black: network of QIF neurons; orange: result of mean-field equations Eq 1) and raster plot of the response for an initial condition in the low-activity state (top, r ≈ 6Hz) and high-activity state (bottom, r ≈ 73Hz), as well as the oscillatory forcing I(t). A At low enough frequencies, the system is pushed from the low- to the high-activity state. B At slightly higher frequencies, both states persist under the forcing. C Driven with forcing from an intermediate range of frequencies, the state with high firing activity destabilizes in favor of the state with low firing activity. D At high frequencies, both states persist under the forcing. Parameters: τ = 20ms, η = −10, Δ = 2, , A = 1.
Fig 2.
Switching behavior at the macroscopic scale.
A Bifurcation analysis of the stationary states identifies a bistable regime for large enough J where a stable focus (red) and a stable node (blue) coexist, separated by a saddle (dotted green). The color-coded curve represents the bifurcation diagram for the value of used here, and the grey curves represent the bifurcation diagrams at different values (left to right: 4J/3, 2J/3, J/3). B The different dynamic regimes of the forced system are shown here as a function of the amplitude A and the frequency f of the forcing. Green: Recall; Red: Clearance; Grey: no switching. Orange: only one globally stable periodic orbit exists due to the system being entrained to the forcing, hysteretically switching between the node and the focus. C Example time traces from B, with initial conditions chosen to be the focus (red) or the node (blue). D The heuristic firing-rate equations Eq 4 show an equivalent fixed point structure, with the exception that the focus is a node here. E Clearance does not occur in the firing-rate equations, as the node cannot be destabilized by nonlinear resonance. Parameters: τ = 20ms, η = −10, Δ = 2,
, A = 1.
Fig 3.
Linear and nonlinear response.
A Linear response of focus, saddle and node to sinusoidal and non-sinusoidal inputs, with the focus showing a characteristic resonant response at approximately 40Hz. The response of the focus to non-sinusoidal input shows additional sub-harmonic resonances. B Nonlinear response of the fixed points by means of bifurcation analysis in the forcing frequency for different amplitudes. Non-sinusoidal forcing leads to a richer bifurcation structure. C Bifurcation diagram in f for non-sinusoidal forcing with A = 1, and comparison with numerical results (bottom). The bifurcation structure is governed by saddle-node bifurcations (SN) and period-doubling bifurcations (PD). Branches of period-doubled solutions are omitted here. D A two-parameter bifurcation analysis of the focus reveals the loci of saddle-node bifurcations (red) and period-doubling bifurcations (blue) in the (f, A)-plane. We compare these with the logarithmic mean squared deviation (log MSD) from the fixed point (color scale), obtained by time simulations. Grey areas indicate regions where the system leaves the basin of attraction of the focus. Parameters: τ = 20ms, η = −10, , Δ = 2.
Fig 4.
Switching in a network of two competing populations of neurons.
A We consider two identical populations with recurrent excitatory connections and mutual inhibition. B Bifurcation diagram of the fixed points of the system. The system can be in a symmetric state (black) or asymmetric state (grey). We choose a point in the tri-stable regime (η = −6, vertical line), where either both populations are quiescent, or one population is active and the other quiescent. The insets show the stable states (two asymmetric, one symmetric). C Applying global forcing with slow frequency (2Hz) does not lead to the activation of either of the asymmetric patterns, due to the lack of symmetry breaking mechanisms. D Driving the system with independent noise sources (zero-mean Ornstein-Uhlenbeck process) with small noise amplitude σ does not lead to reliable switching due to long residence times. E Combining noise with a protocol that generates oscillations of different frequencies over different time intervals leads to the reliable (but random) activation of one of the two asymmetric patterns and switching between these at 2Hz, and the clearing of a sustained pattern at 40Hz. Parameters: η = −6, Δ = 2, , τ = 20ms, A = 2, σ = 0.05.
Fig 5.
Hopfield network with random patterns.
A A network of 100 neural populations is chosen to encode ten patterns with five populations each. The patterns can overlap. B The corresponding connectivity matrix of the network. C We apply an activation/deactivation protocol. The encoded patterns are randomly activated in the presence of slow oscillations (2Hz), sustained in the absence of oscillations (grey, from 60s to 80s), and deactivated in the presence of fast oscillations (40Hz). All populations have independent noise sources (Ornstein-Uhlenbeck) with amplitude σ. D Relative contribution of each pattern to the total activity of the network. Parameters: η = −8, Δ = 2, J = 10, τ = 20ms, A = 5, σ = 0.2.
Fig 6.
Forced switching with oscillations from an E-I network.
A The network is built such that an excitatory population (E1) and an inhibitory population (I) form a circuit that can generate oscillatory output via the excitatory population (E1), which is fed into another excitatory populations (E2). The latter is in the bistable regime. B Bifurcation diagram of the E1-I-circuit in the parameters ηe and ηi. The organizing bifurcations are a pair of saddle-node bifurcations (SN) of the fixed points, and a Hopf branch (H) that connects to one of the saddle-node branches via a Bogdanov-Takens codimension-two point (BT). (Limit cycles are found below the Hopf branch). C Firing rates and raster plots of the population outputs as a result of the parameter tuning. Time traces of population E2 are portrayed for both stable initial conditions (node and focus). By choosing ηe and ηi accordingly, recall (ηe = −4.4, ηi = −18), clearance (ηe = −1, ηi = −5.5) and bistable response (ηe = 0, ηi = −2) can be observed. Other parameters: ,
, Δ = 2, τ = 20ms. Mean current of E2: η = −10.
Fig 7.
A Here we illustrate the interplay between high frequency forcing and a transient stimulus outside the bistable regime. In the absence of oscillations, a 40ms long stimulus with amplitude 6.8 (bar) does not produce sustained activity in the model system, neither do oscillations on their own. However, the combination of oscillations with the transient stimulus leads to sustained high activity, until the oscillations are turned off (arrow). B Loci of the saddle-node bifurcation representing the lower limit of the bistable area, as functions of η and the frequency of the forcing, for different amplitudes of forcing. The choice of parameters in A is indicated by a triangle. Parameters: η = −11.5, Δ = 2, , τ = 20ms, A = 2, f = 80Hz.
Fig 8.
Change of resonant frequency with model parameters.
We compute the (linear) resonant frequency of the saddle for parameter values in the bistable regime (delimited by black lines), specifically where “recall” occurs using non-sinusoidal forcing (delimited by red line). As “clearance” is caused by nonlinear resonance of the focus, the corresponding frequency band is found near the linear resonant frequency. At fixed values of J the resonant frequency varies approximately by a factor of two across the range of values of η. Other parameters: Δ = 2, τ = 20ms.
Fig 9.
Mechanisms of switching: Quasi-stationary response.
A At amplitudes below the critical range no switching occurs (A = 0.7). B Amplitude values within the critical range lead to switching (A = 1). C At amplitudes above the critical range the system undergoes periodic hysteretic switching (A = 1.3). D Bifurcation diagram of stationary states with critical values for saddle-node bifurcations (ηc1, ηc2) and the choice of model parameter (η0). E Normalized non-sinusoidal forcing over one period (A = 1), with minimum and maximum values indicated. Parameters: η = −10, , Δ = 2, τ = 20ms, f = 0.1Hz.
Fig 10.
Mechanisms of switching: Nonlinear resonance.
A Bifurcation diagram of the focus with f as bifurcation parameter at A = 1. B Inset of A, with period-doubling bifurcations (orange dots) and emerging branches of period-doubled solutions shown. The period-doubling cascade gives rise to stable chaos (grey area), which becomes unstable at lower frequencies. C Example time series from B around the area where the chaotic attractor becomes unstable. Parameters: η = −10, , Δ = 2, τ = 20ms.
Fig 11.
The nonlinear resonance is captured in a canonical normal form for a saddle-“focus”.
A Bifurcation diagram of fixed points of this system, giving rise to a stable focus (red) and an unstable saddle (dotted green). B Top: The bifurcation structure of solutions resulting from non-sinusoidal forcing with f as bifurcation parameter (A = 1). The area between vertical bars contains unstable period-doubled solutions, which is evidence of the existence of a chaotic attractor. Bottom: Inverse of the time T that x needs to reach an absolute value of 106, which is evidence that solutions diverge due to the instability of the chaotic orbit. C Comparison of the nonlinear response of the reduced system (left) with the full system (right). Parameters of normal form: μ = 2, a = 0.4. Parameters of full model: η = −10, , Δ = 2, τ = 20ms.