Renewal theory of coupled neuronal pools: Stable states and slow trajectories

A theory is provided to analyze the dynamics of delay-coupled pools of spiking neurons based on stability analysis of stationary firing. Transitions between stable and unstable regimes can be predicted by bifurcation analysis of the underlying integral dynamics. Close to the bifurcation point the network exhibits slowly changing activities and allows for slow collective phenomena like continuous attractors.

Figure 1

Multistability of excitatory-coupled pools. (A) Spike raster plot of one simulated pool with 100 neurons ($W=30$, $I=2$). The thick blue bar on the time axis indicates the time interval at which the neurons received an additional stimulus. (B) Rate $A\left(t\right)$ derived from simulation in (A). Red dashed lines indicate the predicted stationary states, which are (C) the points of intersection between the identity (thin line) and the right hand side of Eq. (12) (thick line). Red discs correspond to the stable states from B. (D) Same as (A) for two pools with $W=\left(\begin{array}{cc}30& -10\\ 10& 30\end{array}\right)$, and $I={(2,1)}^{\mathrm{T}}$. (E) Same as (B) for $\vec{A}\left(t\right)$ from (D). (F) Isoclines for $A_1$ (black) and $A_2$ [green (gray)]. Red discs indicate stationary solutions from (E).

Figure 2

Complex roots. (Top) Real and imaginary part [color (gray) coded] of $\det \widehat{\Lambda }\left(z\right)$ for uncoupled pools $W=0$. Thick black lines indicate respective nullclines. Right: roots of $\det \widehat{\Lambda }\left(z\right)$ (black squares) are determined by the intersection of the nullclines (red for real part, blue for imaginary part). (Bottom) Introducing synaptic self-coupling $W=30$ limits the range of slow real parts to $\mathrm{Re}\left(z\right)<\beta =0.05$. Left: The singularity in $\widehat{\kappa }\left(z\right)$ guarantees a zero crossing of the real part at $\mathrm{Im}\left(z\right)=0$ (the two right-most panels are equivalent to top).

Figure 3

Population oscillations. (A${}_{1-5}$) Population rate from network simulations for five different inhibitory synaptic coupling strengths $\left|W\right|$. (B${}_{1-5}$) Roots (squares) of the feasibility conditions for the weights from $A$. Color (gray) code corresponds to Fig. 2, top right. (C) Empirical firing rate (black dots) and stationary state (red solid line) from the fixed point equation (12). (D) Real part of the root of Eq. (23). (E) Imaginary part (divided by $2\pi$, red) of the root of Eq. (23) and empirical oscillation frequency (black dots).

Figure 4

Line attractor. (A${}_{1-3}$) State spaces for three different $I/E$ ratios (as indicated). Squares indicate stable stationary solutions, discs indicate unstable stationary solutions. Black thick and green thin lines mark the isoclines (below 33 Hz); cf. Fig. 2. Note that axes are logarithmically scaled. (B${}_{1-3}$) Simulations corresponding to the respective situations in (A). Black (upper) and green (lower) traces represent the rates derived from the two individual pools. The blue (middle) trace is the average activity of the system. Gray vertical bars mark the times of the stimuli. The colors (gray) of levels of the horizontal lines correspond to the stationary states from (A). (C),(D) Bifurcation diagrams. (C) Stationary states as a function of the $I/E$ ratio [colors (gray) of levels from (A) and (B). (D) Roots of the feasibility condition from Eq. (23).

Figure 5

Slow transients. (A),(C),(E) Firing rates of three coupled pools from stochastic simulations. Pool 1 received a strong stimulus of amplitude $I_1=6.5$ between 450 and 500 ms. The coupling strength $k$ between the first and the second pool is $k=30$ [in (A)], $k=20$ [in (C)], and $k=15$ [in (E)]. The rates are averaged over 10 ms. (B),(D),(F) Two different projections (blue lines with dots) of the activity variables from the simulations in (A),(C),(E). The arrow in the left graphs indicate the time course. The temporal separation of the dots is 10 ms. Theoretically obtained fixed points are plotted as orange squares (stable) and yellow discs (unstable). The arrows indicate the direction and magnitude of the slow stable and unstable modes. Arrows are drawn black if the imaginary part of the corresponding root of Eq. (23) is zero. Red (gray) arrows indicate oscillatory modes with nonzero imaginary part of the root. Here, the refractory time is $\tau =10$ ms, and the coupling kernel has a time constant of $1/\beta =30$ ms.