Fig 1.
From microscopic to mesoscopic population dynamics.
(A) Network with microscopic short-term plasticity. Dashed region shows a zoom into a pair of interconnected neurons: presynaptic neuron 1 sends out an unmodulated spike train to postsynaptic neuron 2 that receives the spike-train modulated by short-term depression. (B) Mesoscopic mean-field model with one effective synapse undergoing short-term depression.
Fig 2.
Population spikes in excitatory populations of finite size.
(A) Phase-plane analysis of the macroscopic model (Eq (3) for N → ∞) reveals the backbone of the metastable dynamics due to the proximity of a separatrix (red-dashed) near the unique stable fixed point (red dot = cross-section of the black-dashed nullclines). Trajectories (blue) of the mesoscopic model reproduce population spikes by following the unstable manifold (orange dotted line) of the saddle fixed point (orange diamond). Population spikes have variable amplitude and inter-population spike intervals (ISI), see also (B,C). (D) The mesoscopic models with hybrid noise (jump-diffusion model; blue) and Gaussian noise (diffusion model; orange) accurately capture finite-size fluctuations in the input potential h—note the logarithmic y-scale—and population spikes of the microscopic network dynamics (black) of N = 30 neurons. (E) Power spectra of the input potential h and (F) ISI distributions coincide for all three models. (G-H) same as (D-F) for N = 200. Statistics are for simulations of length Tsim = 100′000s. Model parameters can be found in Table 1.
Fig 3.
Up-down dynamics due to finite-size fluctuations.
Mesoscopic model reproduces noisy bistable population dynamics. (A) Phase-plane analysis of macroscopic dynamics (Eq (3) for N → ∞) reveals two stable fixed points (red): a high-activity focus representing the Up, and the low-activity node the Down state of the system. From the saddle fixed point (orange diamond), an unstable (orange dotted line) and a stable manifold (red dashed line) emerge. The latter acts as a separatrix—trajectories (blue curve) starting from above make an excursion around the unstable limit cycle (green dashed) and converge towards the down state. Finite-size fluctuations can make the trajectory cross the limit cycle into the basin of attraction of the Up state. (B,C) Stochastic trajectory of the mesoscopic dynamics (3) with N = 100 transitioning between Down and Up states. (D) The mesoscopic models with hybrid noise (jump-diffusion model; blue) and Gaussian noise (diffusion model; orange) qualitatively capture Up-Down-dynamics of the microscopic network (black). (E) Power spectrum and (F) histogram of input potential h over simulation of length Tsim = 100′000s. Model parameters can be found in Table 1.
Fig 4.
Hippocampal replay in micro- and mesoscopic ring-attractor network model.
(A) Ring-attractor model of M population units of N LNP spiking neurons with STD. Synaptic weights Jα,β are excitatory for units with nearby place field positions θα and inhibitory at longer distances, see the coupling function on the right. (B) Mesoscopic and (C) microscopic network simulations reveal (i) spontaneous bursts of the averaged activity, resembling SWs, during which replay patterns evolve as (ii) metastable traveling waves, or nonlocal replay events (NLE), along the circular environment—the expected activity rj = f(hj) at location θj = 2πj/M, j = 1, …, M is color-coded. Statistics of the (D) mesoscopic and (E) microscopic simulations perfectly match each other with respect to: (i) the distribution of event duration, (ii) the correlation between the number of peaks per burst and its duration, (iii) the correlation between the length of the traveled path during an event and its duration (the red curves in panels (ii,iii) are linear regression curves), as well as (iv) the distribution of average bump speed during an event, computed from events with more than one peak; see Methods for more details, Table 1 for model parameters and Table 2 for simulation results.
Fig 5.
Comparison of fluctuation-induced and depression-induced hippocampal replay dynamics.
(A) Interburst-interval distributions and (B) serial correlations of the event speeds of consecutive NLEs for (i) the meso- and microscopic models and (ii) the deterministic models (green: original Romani-Tsodyks model [21], purple: macroscopic model in the fatigue-driven regime obtained by setting Nα → ∞ and μ = −0.9 (instead of μ = −1.4)). (C) On shorter time scales, all models capture the discontinuous nature of the replayed trajectory: (i) Single NLE of the mesoscopic model. (ii) Place field positions decoded using PVA in bins of 50ms (black dots). This “replayed trajectory” deviates from a straight line (red-dashed) corresponding to a hypothetical uniform motion. (iii) Increments of the movement trajectory exhibit strongly irregular movement features (black dots) in contrast to the constant increments expected for a straight line (red-dashed). (D) (i): The distributions of reconstructed step-sizes of the meso- (blue) and microscopic models (gray histogram) coincide and strongly deviate from the narrow distributions of average step sizes (computed for each NLE by fitting straight lines to individual movement trajectories) for the meso- (red) and microscopic models (gray thin line). The average step sizes vary for different NLE’s, causing the non-zero width of their distribution. (ii) Similar behavior is observed for the macroscopic model (purple; with increased external input μ = −1.4 ↦ −0.9) and the Romani-Tsodyks (RT) model (green histogram); red/gray thin lines correspond to average step sizes in the macro/RT-models, respectively. See main text and Methods for more details.
Fig 6.
Spontaneous replay switches between multiple environments.
In the (A) mesoscopic and (B) microscopic ring-attractor network storing multiple environments, metastable replay dynamics spontaneously emerge due to finite-size fluctuations when decreasing the population size from N = 500 (orange/red in panels i) to N = 50 (blue/black) per unit. Nonlocal replay events (NLEs) occur randomly in exclusively one of three environments, while activity in the respective other two is suppressed. The resulting activation sequences of replayed environments—in (A) the activation sequence reads 313123122312321—are analyzed with respect to (C) the transition probabilities between subsequently active environments and (D) sequential activation patterns. In the meso- and microscopic models, transitions from environment k to j are equally likely for all pairs (k, j) ∈ {1, 2, 3}2. But the deterministic (macro and RT) models show a clear preference for transitions 1 → 3 → 2 → 1, which is also apparent in the high probability of the corresponding subsequences of three distinct, subsequently active environments. (E) Larger heterogeneity with respect to NLE duration and interburst intervals (IBI), as assessed by the respective means μ and coefficients of variation CV, further distinguishes the more variable metastable regime of the micro-/mesoscopic vis-à-vis macroscopic/deterministic models. Model parameters can be found in Table 1.
Table 1.
Table 2.
Table 3.
Simulation results complementing Fig 6.