Working memory dynamics and spontaneous activity in a flip-flop oscillations network model with a Milnor attractor

Abstract

Many cognitive tasks require the ability to maintain and manipulate simultaneously several chunks of information. Numerous neurobiological observations have reported that this ability, known as the working memory, is associated with both a slow oscillation (leading to the up and down states) and the presence of the theta rhythm. Furthermore, during resting state, the spontaneous activity of the cortex exhibits exquisite spatiotemporal patterns sharing similar features with the ones observed during specific memory tasks. Here to enlighten neural implication of working memory under these complicated dynamics, we propose a phenomenological network model with biologically plausible neural dynamics and recurrent connections. Each unit embeds an internal oscillation at the theta rhythm which can be triggered during up-state of the membrane potential. As a result, the resting state of a single unit is no longer a classical fixed point attractor but rather the Milnor attractor, and multiple oscillations appear in the dynamics of a coupled system. In conclusion, the interplay between the up and down states and theta rhythm endows high potential in working memory operation associated with complexity in spontaneous activities.

Fig. 1

Fixed points analyses for one cell. a Cylinder space (S,ϕ) with nullclines (orange for dS/dt = 0, yellow for dϕ/dt) and some trajectories. Left to right shows the three possible scenario: M₀ is stable fixed-point for μ < μ_c, M₀ is the Milnor attractor for μ = μ_c and M₀ is unstable fixed-point for μ > μ_c. b Evolution of the two Lyapunov exponents of the system at M₀ (in blue) and M₁ (in green) in function of σ (ρ = 1). As expected, one exponent becomes null at σ = μ_c. c Evolution of the two fixed points, M₀ and M₁, when σ is varied. μ = μ_c corresponds to a transcritic bifurcation

Fig. 2

Normal form reduction near μ_c. The two nullclines and appear. Plain arrows indicate the direction of the trajectories for x₂ = 0

Fig. 3

Evolution of the maximum and minimum values of S, and of the dominant frequency obtained by FFT when the input current is varied

Fig. 4

Voltage response to input currents. a for current steps, b for oscillatory input of increasing frequency. Upper figures show the input pattern. Then, from up to down, the voltage response is shown for our complex unit, then for a unit without phase modulation, finally, when only the phasic output is considered

Fig. 5

Extrema of the membrane potential during the up-state ( and ) and normalized phase difference () for a two-unit network when either the internal coupling strength (a) or the synaptic weight (b) is varied. Lower figures demonstrate the presence of complex dynamics in specific region of the upper figures by showing the bifurcation diagram for the local minima of the membrane potential. For (a), w = 0.75. For (b), σ = 0.9

Fig. 6

S_i temporal evolution, (S₁,S₂) phase plane and (S_i, ϕ_i) cylinder space. a Up-state oscillation for strong coupling. b Multiple frequency oscillation for intermediate coupling. c Down-state oscillation for weak coupling

Fig. 7

Spontaneous activity in a 80 units network containing eight overlapped cell assemblies of 10 cells each. The upper and the middle figures show respectively the membrane potential and a rasterplot of the activity of each individual cell (one color per cell). The lower figure shows the reactivation of the different cell assemblies (each assembly has its own color and letter). Periods of no-activity alternate with periods of activity during which specific cell assemblies are preferentially activated

Fig. 8

Working memory in a 80 units network containing eight overlapped cell assemblies of 10 cells each. In both a and b, upper figures show the membrane potential of each individual cell (each cell one color). Middle figures show rasterplots of individual cells activity (a cell is said to be in an active state if its spike density is larger than 0.5). Lower figures show the reactivation of the different cell assemblies (one color and letter per cell assembly). a An external stimulus was impinging part (40%) of one cell assembly during a short transient (10 computational time steps). As a result, this cell assembly is continuously activated as a short term memory. b External stimuli are successively applied to part (40%) of three cell assemblies (each CA is stimulated during 100 computational time steps; i.e. approximately 200 ms). After stimulation, we observe that these three cell assemblies have sustained activity