Table 1.
Exemplary list of dynamical systems analysis software packages that are supported by one of the backends available in PyRates.
Fig 1.
(A) Depiction of the user interface: PyRates models are implemented via different templates that can be defined via a YAML or Python interface. OperatorTemplate instances are used to define equations and variables and serve as basic building blocks for NodeTemplate and EdgeTemplate instances. The latter can be used to define CircuitTemplate instances which are used to represent the final models in PyRates. CircuitTemplate instances can also be incorporated in higher-level CircuitTemplate instances to allow for complex hierarchies, as depicted by the coupling of circuit 1 and circuit 2 within a CircuitTemplate. (B) Structure of the backend: Each model is translated into a compute graph, which in turn is parsed into a backend-specific model implementation. The latter can be used for code generation and numerical analyses.
Fig 2.
Entrainment of the Van der Pol oscillator in response to periodic forcing.
(A) Coherence between the state variables x of the Van der Pol oscillator (VPO) and θ of the Kuramoto oscillar (KO). For each pair of ω and J, we bandpass-filtered x at the frequency ω and extracted the phase of the bandpass-filtered signal via the Hilbert transform. We then created a sinusoidal signal from the VPO and KO phases and used scipy.signal.coherence to calculate the coherence between the two sinusoids. The result is depicted as color-coding. (B and C) State variables x (black) and θ (orange) displayed over time. (B) No entrainment of the VPO phase for ω = 0.33 and J = 0.5. (C) Entrainment of the VPO phase to the KO phase for ω = 0.42 and J = 1.0.
Fig 3.
Bifurcation analysis of the QIF model.
(A) Bifurcation diagram showing the solutions of Eqs (6)–(9) in the state variable r as a function of the parameter . Solid (dotted) lines represent stable (unstable) solutions. Bifurcation points are depicted as symbols along the solution branches. Green circles represent Hopf bifurcations whereas grey triangles represent fold bifurcations. (B) Convergence of the average firing rate r of the QIF model to a steady-state solution in the asynchronous, high-activity regime (
). (C) Convergence of the average firing rate r of the QIF model to a periodic solution in the synchronous, oscillatory regime (
).
Fig 4.
Comparison of the dynamics of the target leaky integrator model and the fitted leaky integrator model.
(A) Logarithm of the mean-squared error (color-coded), depicted over the search range of the two parameters that were optimized: κ and τ. The white trace shows the steps taken by the optimizer from its initialization point to the global minimum. (B and C) Rate signals of all N LI units over time of the fitted network and the target network, respectively.