Fig 1.
The largest organizational unit of a network model is the Circuit. Any circuit may also consist of multiple hierarchical layers of subcircuits. Panel (A) depicts an imaginary circuit that encompasses four subcircuits that represent one brain region each. One of these local circuits is a Jansen-Rit circuit (B), consisting of three neural populations (PC, EIN, IIN) and the connections between them. One node (C) may consist of multiple operators containing the mathematical equations. Here, two rate-to-potential operators (RPO) convolute incoming firing rates with an alpha kernel to produce post-synaptic potentials. These are summed into a combined membrane potential V. The potential-to-rate operator (PRO) transforms V into an outgoing firing rate mout via a sigmoid function. Inset graphs give a qualitative representation of the operators and evolution of the membrane potential. Edges (lines in A and B) represent information transfer between nodes. As panel (D) shows, edges may also contain operators. By default, edges apply a multiplicative weighting constant and can optionally delay the information passage with respect to time. The equation shown in panel (D) depicts this default behavior.
Fig 2.
PyRates is separated into frontend, intermediate representation (IR) and backend. The frontend features a set of interfaces to define network models. These are then translated into a standardized structure, called the IR. Simulations are realized via the backend, which transforms the high-level IR into lower-level representations for efficient computations. The frontend can easily be extended with new interfaces, while the backend can be swapped out to target a different computation framework.
Fig 3.
Jansen-Rit and Montbrió model validations.
A Shows the simulation results obtained from a single Jansen-Rit model. On the left hand side, the average membrane potentials of the pyramidal cell population are depicted for different connectivity scalings C. On the right hand side, the dominant oscillation frequency of the pyramidal cell membrane potentials (evaluated over a simulation period of 60 seconds) is depicted for different synaptic time-scales τe and τi. The frequencies are categorized into the following bands: δ (1-4 Hz), θ (4-8 Hz), α (8-12 Hz), β (12–30 Hz), γ (> 30 Hz) and h.s. (hyper signal) for signals not representative of any EEG component. B Shows the simulation results obtained from a single Montbrió model. The average membrane potentials v, average firing rates r and input currents are depicted for constant and oscillatory input on the left and right hand side, respectively. Time-dependent variables are reported in units of τ, which was set to τ = 1.0 in accordance with the simulations performed by Montbrió and colleagues. Following the definitions of Montbrió and colleagues, membrane potential and input are reported as unit-less variables.
Fig 4.
Benchmark results for 1 s simulations run in PyRates with a simulation step-size of 0.1 ms. A and B Show average simulation durations over 10 independent simulations for networks with different numbers of Jansen-Rit circuits (N) and differently dense coupling between the JRCs (p), performed on the NumPy (CPU) and tensorflow (CPU+GPU) backend, respectively. C Shows the average simulation durations for parameter sweeps over N different parametrizations of a network of 2 bidirectionally, delay-coupled Jansen-Rit circuits. Averages were again calculated over 10 independent runs of each parameter sweep.