Models

Neural mass and field models in the ontology, plus models shared by the community.

Hamiltonian antidot lattice (periodic Sinai-like billiard) with smooth circular repulsive potentials and perpendicular magnetic field. Models guiding center dynamics and magnetotransport phenomena in mesoscopic systems. Smooth radial repulsion implemented as a quartic polynomial potential inside radius (d0/2 + c) matching the original implementation (Datseris 2019); outside region force is zero.

ArnoldCatMap

Arnold’s cat map: linear automorphism of the torus with hyperbolic dynamics and uniform stretching and folding.

BetaTransformationMap

The β-transformation (generalized Bernoulli map): x_{n+1} = (β x_n) mod 1. For β=2 it is the dyadic map with symbolic dynamics given by binary shift.

CakanObermayer

Adaptive linear-nonlinear (aLN) mean-field model of a network of adaptive exponential integrate-and-fire (AdEx) neurons, comprising one excitatory (E) and one inhibitory (I) population. Originally developed by Augustin, Ladenbauer, Liebe & Obermayer (2017) and used as the whole-brain mean-field in neurolib by Cakan, Jajcay & Obermayer (2021). The legacy tvb-o.owl ontology referred to this model by the short name ``Cortical`` (suffix ``_CO``); ``iri`` is preserved for backward ontology mapping while ``name`` reflects the canonical attribution. Defaults are taken verbatim from ``neurolib/models/aln/loadDefaultParams.py``. Note that the full aLN dynamics require a precomputed transfer-function lookup table (``aln-precalc/quantities_cascade.h5``) that maps the filtered mean input ``mu`` and the membrane-potential standard deviation ``sigma`` to a steady-state firing rate. The state-variable equations below are the structural filtered-mean-input scaffold; the nonlinear transfer function ``r_E = Phi(mu_E, sigma_E)`` cannot be expressed in YAML and is therefore approximated as a sigmoid for backend code generation.

Chua

This is a 3D continuous system that exhibits chaos. Chua designed an electronic circuit with the expressed goal of exhibiting chaotic motion, and this system is obtained by rescaling the circuit units to simplify the form of the equation. [^Chua1992] The parameters are a, b, m0, and m1. Setting a = 15.6, m0 = -8/7 and m1 = -5/7, and varying the parameter b from b = 25 to b = 51, one observes a classic period-doubling bifurcation route to chaos. [^Chua2007]

CoombesByrne

4D model describing the Ott-Antonsen reduction of infinite all-to-all coupled QIF neurons (Theta-neurons). Note: the original equations describe the dynamics of the Kuramoto parameter $Z$. Using the conformal transformation $Z=(1-W^\star)/(1+W^\star)$ and $W= \pi r + i V$, we express the system dynamics in terms of two state variables $r$ and $V$ representing the average firing rate and the average membrane potential of our QIF neurons. The conductance variable and its derivative are $g$ and $q$.

CoombesByrne2D

No description.

CoupledRoessler

Pair of diffusively coupled Rössler oscillators used to study phase and generalized synchronization, lag synchronization and coupling asymmetry effects.

CoupledStandardMaps3

Three coupled Chirikov standard maps with nearest-neighbor coupling on a ring. This explicit unrolled form avoids symbolic indices by naming each (θ_i, p_i) pair.

DecoBalancedExcInh

Deco Balanced Exc-Inh model. It extends the Reduced Wong-Wang Exc-Inh (RWW_EI) dynamic mean-field model with an effective regional gain parameter M_i that scales both the excitatory and inhibitory input currents, capturing the dynamical consequences of regional heterogeneity in the brain's transcriptional landscape (Deco et al., 2021).

DoublePendulum

Famous chaotic double pendulum (planar) with two point masses connected by massless rods. The ordering of variables is [θ₁, ω₁, θ₂, ω₂]. Shows sensitive dependence and rich energy-dependent phase space structure. The auxiliary variables φ = θ₂ - θ₁ and Δ = (M1 + M2) - M2 cos² φ appear in the equations.

DuffingForced

Forced Duffing oscillator with linear damping and periodic forcing. Exhibits rich nonlinear dynamics including bifurcations, quasiperiodicity, and chaos depending on forcing amplitude and frequency.

DumontGutkin

No description.

Epileptor

The Epileptor is a composite neural mass model of six dimensions which has been crafted to model the phenomenology of epileptic seizures. Equations and default parameters are taken from Jirsa et al. (2014).

Epileptor2D

Epileptor2D (E2D) is a phenomenological neural mass model consisting in the two-dimensional reduction ('x', 'z') of the original Epileptor model (see Epileptor5D; Proix et al., 2014, 2017). Note: ------ - Equations and default parameters are taken from (Proix et al.,2014), - The slow permittivity state-variable (z_E2D) can be modified to account for the time difference between the interictal (between seizures) and ictal (during seizure) states (see Proix et al., 2014).

Epileptor3DStefanescuMcDonald

Stefanescu-McDonald reduced 3D Epileptor variant migrated as a placeholder scaffold from the legacy tvb-o.owl ontology. The original publication for this exact variant could not be located; the only OWL-side parameter is `N` (population size, grounded to "regulation of resting membrane potential"). Equations are minimal placeholders: a fast (x1) - slow (z) excitable system inspired by the canonical Epileptor (Jirsa et al. 2014), gated by N. This scaffold preserves the ontology lineage and parameter grounding; it should be replaced with the published equations once located, or removed if the variant is confirmed obsolete.

Epileptor5D

Epilepor5D (E5D) is a phenomenological, coupled, nonlinear five-dimensional (i.e., five state-variables ('x1', 'y1', 'z', 'x2', 'y2')) neural mass model able to realistically reproduce the temporal dynamics of epileptic seizures and the alternating sequence of seizures (ictal and interictal state; Jirsa et al.,2014; El Houssaini et al., 2015, 2020). Epileptor5D comprises three different time scales interacting together and accounting for various electrographic patterns: - the fastest and intermediate time scales are two coupled oscillators ((x1, y1) and (x2, y2)), accounting respectively for the low-voltage fast discharges (i.e., very fast oscillations) and spike-and-wave discharges. - the slowest time scale is responsible for leading the autonomous switch between interictal and ictal states and is driven by a slow-permittivity variable z. This switching is accompanied by a direct current (DC) shift that has been recorded in vitro and in vivo. The main output of the model: -x1+ x2, bears analogy with the field potential, while the precise biophysical equivalent of the z variable is unknown and will be likely complex. Note: ------ - Equations and default parameters are taken from (Jirsa et al.,2014 & El Houssaini et al., 2015), - The integral coupling function g(x1) can be rewritten as an ordinary differential equation, which is technically introduced, here, as a sixth state-variable (see Jirsa et al.,2014), - The slow permittivity state-variable (z_E5D) can be modified to account for the time difference between the interictal (between seizures) and ictal (during seizure) states (see Proix et al., 2014).

EpileptorCodim3

The Epileptor codim 3 model is a neural mass model containing two subsystems acting at different timescales. The fast subsystem (x, y) uses the unfolding of a degenerate Takens-Bogdanov bifurcation of codimension 3; the slow subsystem (z) steers the fast one back and forth along a great arc of a sphere, producing bursting behavior. It reproduces almost all classes of bursting predicted for systems with a planar fast subsystem. The default parameters correspond to Hysteresis-Loop burster class c2s (Saddle-Node at onset, Saddle-Homoclinic at offset, branch N=1, with the stabilising modification for negative dstar).

EpileptorCodim3SlowMod

The Epileptor Codim 3 model with ultra-slow modulation of bursting classes (Saggio et al., 2017). It is a neural mass model containing a fast subsystem (x, y) built from the unfolding of a degenerate Takens-Bogdanov bifurcation of codimension 3 and a slow subsystem (z) that steers the fast one back and forth along a path in parameter space, producing bursting. Two additional ultra-slow state variables (uA, uB) modulate the offset (A) and onset (B) points of the burster along great arcs on a sphere of radius R, allowing the model to switch between different classes of bursters (Hysteresis-Loop bursters c0, c0', c2s, c3s, c4s, c10s, c11s, c2b, c4b, c8b, c14b, c16b as classified in Saggio et al. 2017 Table 2). The bifurcation parameters (mu1, mu2, nu) at the offset (A) and onset (B) points are obtained by parametrising great arcs between the points Ain/Aend and Bin/Bend on the sphere. This capture uses the default branch of the resting state (N = 1) and the default stabilising modification (modification = True).

EpileptorRestingState

Epileptor Resting-State (ERS) is an extension of the phenomenological neural mass model of Epileptor5D, tuned to express regionally specific physiological oscillations in addition to the epileptiform discharges (Courtiol et al., 2020). This extension was made using the Generic 2-dimensional Oscillator model (parametrized close to a supercritical Hopf Bifurcation) (Sanz-Leon et al., 2013, 2015) to reproduce the spontaneous local field potential-like signal. This model, its motivation and derivation can be found in the published article (Courtiol et al., 2020).

FitzHughNagumo

Famous excitable system which emulates the firing of a neuron, reducing the biophysical Hodgkin-Huxley description to two variables while preserving excitability and recovery dynamics. Captures threshold behavior, refractoriness and sustained spiking under drive.

FitzHughNagumo1969

No description.

ForcedPendulum

Driven damped pendulum showing classic routes to chaos (period doubling and intermittency) and rich phase locking structure in parameter space.

GastSchmidtKnosche_SD

4D model describing the Ott-Antonsen reduction of infinite all-to-all coupled QIF neurons (Theta-neurons) with Synaptic Depression adaptation mechanisms [Gastetal_2020]_. The two state variables :math:`r` and :math:`V` represent the average firing rate and the average membrane potential of our QIF neurons. :math:`A` and :math:`B` are respectively the adaptation variable and its derivative. The equations of the infinite QIF 2D population model read .. math:: \dot{r} &= 1/\tau (\Delta/(\pi \tau) + 2 V r)\\ \dot{V} &= 1/\tau (V^2 - \tau^2 \pi^2 r^2 + \eta + J \tau r (1 - A) + I)\\ \dot{A} &= 1/\tau_A (B)\\ \dot{B} &= 1/\tau_A (-2 B - A + \alpha r) \\ .. [Gastetal_2020] Gast, R., Schmidt, H., & Knösche, T. R. (2020). A mean-field description of bursting dynamics in spiking neural networks with short-term adaptation. *Neural Computation*, 32(9), 1615-1634.

GastSchmidtKnosche_SF

No description.

Generic2dOscillator

The Generic 2-Dimensional Oscillator (G2D) is a phenomenological, coupled, nonlinear two-dimensional (i.e., two state-variables ('V', 'W')) oscillatory, neural mass model. The G2D is a generalization of the well-known FitzHugh-Nagumo model (FitzHugh, 1961; Nagumo et. al, 1962), adapted here for reproducing a wilder class of dynamical configurations of physiological phenomena as observed in neuronal population using phase-portrait method.

GenericLinear

The Generic Linear (GL) is a phenomenological, linear one-dimensional (i.e., one state-variable ('x')) neural mass model (Galán, 2008).

Gissinger

Low-order model for geomagnetic field reversals exhibiting chaotic polarity changes. Captures interplay between dipole (Q), quadrupole (D) and velocity (V) modes.

GrebogiMap

Two-dimensional map with chaotic transients and fractal basin boundaries; converges to one of two fixed points depending on initial condition.

GuckenheimerHolmes

Symmetric three-dimensional normal form near a triple zero eigenvalue illustrating mode interactions, heteroclinic cycles, and the onset of complicated dynamics in equivariant bifurcation problems. Tunable linear and quadratic interaction terms produce transitions between steady states, periodic orbits, and chaotic transients.

HH_KineticScheme

Cell with HH Na channel and kinetic-scheme K channel, matching NeuroML2 Ex4.

HH_Tissue_Q10

No description.

Halvorsen

Smooth three-dimensional chaotic flow with rotational symmetry and robust strange attractor arising from quadratic cross-couplings and constant drive. Exhibits intertwined scroll-like structure and sensitive dependence useful for benchmarking nonlinear state estimation and control strategies.

HenonHeiles

The Hénon–Heiles system is a conservative 2 DoF Hamiltonian model introduced to study stellar motion near a galactic center and the search for a third integral of motion. The default initial condition is a typical chaotic orbit.

HenonMap

The Hénon map is a two-dimensional discrete-time dynamical system exhibiting a strange attractor for the default parameters a=1.4, b=0.3. It serves as a simple model for chaotic dynamics in smooth invertible maps.

HindmarshRose

The Hindmarsh-Rose model reproduces the bursting behavior of a neuron's membrane potential, characterized by a fast sequence of spikes followed by a quiescent period. The x variable is the membrane potential; y the fast recovery (ionic current) variable; z a slow adaptation current. Parameter sets modulate transitions between quiescence, tonic spiking and bursting.

HodgkinHuxley

Canonical conductance-based neuron model capturing the biophysical basis of the action potential via voltage-gated sodium and potassium currents and a passive leak. Fast activation/inactivation (m,h) of sodium and slower potassium activation (n) shape spike initiation, refractory period, and repetitive firing under sustained current injection.

HodgkinHuxley_Q10

No description.

Hopfield

The Hopfield neural network is a discrete time dynamical system composed of multiple binary nodes, with a connectivity matrix built from a predetermined set of patterns. The update, inspired from the spin-glass model (used to describe magnetic properties of dilute alloys), is based on a random scanning of every node. The existence of a fixed point dynamics is guaranteed by a Lyapunov function. The Hopfield network is expected to have those multiple patterns as attractors (multistable dynamical system). When the initial conditions are close to one of the 'learned' patterns, the dynamical system is expected to relax on the corresponding attractor. A possible output of the system is the final attractive state (interpreted as an associative memory). Various extensions of the initial model have been proposed, among which a noiseless and continuous version [Hopfield 1984] having a slightly different Lyapunov function, but essentially the same dynamical properties, with more straightforward physiological Interpretation. A continuous Hopfield neural network (with a sigmoid transfer function) can indeed be interpreted as a network of neural masses with every node corresponding to the mean field activity of a local brain region, with many bridges with the Wilson Cowan model [WC_1972]. Note: - This model uses the modifications implemented by Golos et al. (2015).

HyperRoessler

Four-dimensional hyperchaotic extension of the Rössler system introducing an additional variable that yields a second positive Lyapunov exponent over wide parameter ranges while preserving the spiral-type attractor structure.

IaFCell

No description.

IkedaMap

Ikeda map modeling light propagation in a ring cavity with nonlinear phase; exhibits rich strange attractors.

IntegrateAndFire

No description.

Izhikevich2007

Izhikevich 2007 model

Izhikevich2007Cell

No description.

IzhikevichBurst

No description.

IzhikevichCell

No description.

JansenRit

The Jansen-Rit is a neurophysiologically-inspired neural mass model of a cortical column (or area), developed to simulate the electrical brain activity, i.e., the electroencephalogram (EEG), and evoked-potentials (EPs; Jansen et al., 1993; Jansen & Rit, 1995). It is a 6-dimensional, non-linear, model describing the local average states of three interconnected neural populations: pyramidal cells (PCs), excitatory and inhibitory interneurons (EINs and IINs), interacting through positive and negative feedback loops. The main output of the model is the average membrane potential of the pyramidal cell population, as the sum of the potential of these cells is thought to be the source of the potential recorded in the EEG.

Jansen-Rit Neural Mass Model (1995)

Lumped-parameter cortical column model with three neural populations: pyramidal cells, excitatory interneurons, and inhibitory interneurons. Each population is modeled by a PSP block (2nd-order linear filter) and a sigmoidal potential-to-firing-rate transformation. The model produces EEG-like output as the pyramidal PSP difference v_pyr = y1 - y2. Based on Lopes da Silva et al. (1974, 1976) and Jansen et al. (1993).

KIonEx

It describes the mean-field activity of a population of Hodgkin-Huxley-type neurons (Depannemaker et al 2022) linking the slow fluctuations of intra- and extra-cellular potassium ion concentrations to the mean membrane potential, and the synaptic input to the population firing rate. The model is derived as the mathematical limit of an infinite number of all-to-all coupled neurons, resulting in 5 state variables: :math:`x` represents a phenomenological variable connected to the firing rate, :math:`V` represent the average membrane potential, :math:`n` represents the gating variable for potassium K, :math:`\Delta K_{int}` represent the intracellular potassium concentration, :math:`K_g` represents the extracellular potassium buffering by the external bath """

Kuramoto

The Kuramoto model is a model of synchronization phenomena derived by Yoshiki Kuramoto in 1975 which has since been applied to diverse domains including the study of neuronal oscillations and synchronization.

KuramotoModel2

No description.

Labyrinth

Labyrinth attractor: b = 0 limiting case of Thomas' cyclically symmetric system, producing conservative-like intricate winding trajectories through a trigonometric vector field.

LarterBreakspear

The Larter-Breakspear is an extension (Breakspear et al., 2003a, 2003b) of the biophysical-inspired neural mass model of a cortical column (or area) from Larter et al. (1999), initially developed to simulate firing rate activity from focal region involved in partial seizure. It is determined by voltage- and ligand-gated ions channels and feedback between intensively interconnected excitatory and inhibitory neurons. The Larter-Breakspear is a 3D model describing the local average states of two interconnected neural populations: pyramidal cells (PCs) and inhibitory interneurons (IINs), with an additional variable representing the potassium channels in the population of PCs. The membrane potential of the pyramidal cells is the focus of the model and is governed by sodium, potassium, calcium and “leaky” ion channels, of which the voltage-gated potassium channels are modelled in more detail. The excitatory to excitatory connections are modelled in more detail as glutamatergic connections with AMPA and NMDA receptors. Note: - Equations and default parameters are taken from (Breakspear et al., 2003b), - All equations and parameters are non-dimensional and normalized to neural capacitance C = 1.

Linear

No description.

LogisticMap

One-dimensional logistic map x_{n+1} = r x_n (1 - x_n), archetypal model for routes to chaos via period-doubling. Default r=4.0 yields fully developed chaos.

Lorenz63

The famous three dimensional system due to Lorenz [^Lorenz1963], shown to exhibit so-called "deterministic nonperiodic flow". It was originally invented to study a simplified form of atmospheric convection. Currently, it is most famous for its strange attractor (occuring at the default parameters), which resembles a butterfly. For the same reason it is also associated with the term "butterfly effect" (a term which Lorenz himself disliked) even though the effect applies generally to dynamical systems. Default values are the ones used in the original paper.

Lorenz84

Low-order atmospheric circulation model exhibiting multistability and fractal basin boundaries with coexisting attractors under standard parameter set.

Lorenz96

Ten-dimensional explicit expansion of the Lorenz-96 system: dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F with cyclic indices modulo N (here N=10 unrolled to avoid symbolic indexing that the current parser cannot interpret). Initial conditions follow the Julia reference default u0 = range(0; length=N, step=0.1) i.e. x_i(0) = 0.1*i.

LorenzBounded

Lorenz system modified by smooth radial confinement factor f = 1 - (||X||^2)/r^2 producing bounded dynamics with altered basin geometry while retaining core stretching and folding mechanisms.

LorenzDiffusionless

Diffusionless variant of Lorenz-type flow supporting a pair of linked Malasoma strange attractors with fractal basin boundaries for typical parameter values.

LotkaVolterraPredPrey

The famous Lotka-Volterra model is a simple ecological model describing the interaction between a predator and a prey species (or also parasite and host species). It has been used independently in epidemics, ecology, and economics. The prey (x) grows exponentially in absence of predators; predation converts prey into predator growth; predator (y) declines without prey. Default parameters produce sustained oscillations.

MagneticPendulum

Damped planar magnetic pendulum over N fixed magnets forming a symmetric multistable potential. Exhibits fractal basin boundaries and sensitive dependence in final magnet attraction region selection.

MannevilleSimpleMap

Simple intermittency map x_{n+1} = [(1+ε)x_n + (1-ε)x_n^2] mod 1, used to illustrate bursts separated by quiescent phases.

Jansen-Rit Neural Mass Model (single column)

JR model with canonical parameters; long-range input is injected via a coupling term. Output is pyramidal potential difference.

MontbrioPazoRoxin

No description.

MoreChaosExample

Three-dimensional chaotic flow whose attractor is multifractal with dimension close to 3, introduced as an additional example in discussions about the need for new chaos exemplars.

MorrisLecar

Two-variable conductance-based neuron model capturing excitability via fast calcium and slower potassium dynamics; a simplification of Hodgkin-Huxley retaining rich bifurcation structure.

NLDCoupledLogisticMaps4

Four-dimensional unrolled version of the nonlinear diffusion-like coupled logistic maps with all-to-all coupling. General D requires symbolic indexing not supported by the current parser, hence D=4 explicit form.

NoseHoover

Nosé–Hoover thermostat flow: minimal three-dimensional conservative system with time-reversible chaotic dynamics arising in statistical mechanics and later rediscovered as an algebraically simple chaotic flow.

PinskyRinzelCA3

No description.

PomeauMannevilleMap

Intermittent map with laminar phases and chaotic bursts; here given in the symmetric piecewise form commonly used in the literature.

QuadrupoleBosonHamiltonian

Conservative Hamiltonian system modeling quadrupole surface vibrations in a nuclear context. Adds a quartic term to Henon–Heiles-like structure producing energy-dependent chaoticity. Default initial condition is chaotic.

ReducedSetFitzHughNagumo

Reduced set of FitzHugh-Nagumo oscillators (Stefanescu & Jirsa 2008); 3 modes coupled via mode-mixing matrices with Gaussian-quadrature coefficients recomputed from mu/sigma.

ReducedSetHindmarshRose

Reduced set of Hindmarsh-Rose oscillators (Stefanescu & Jirsa 2008); 3 modes coupled via mode-mixing matrices with Gaussian-quadrature coefficients recomputed from mu/sigma.

ReducedWongWang

Reduced WongWang (RWW) is a biologically-inspired one-dimensional (i.e., only one state-variable 'S') neural mass model that approximates the realistic temporal dynamics of a detailed spiking and conductance-based synaptic large-scale network (Deco et al., 2013). RWW is the dynamical mean-field (DMF) reduction of the Reduced WongWang Exc-Inh model, that consists in disentangling the contribution of the two neuronal populations (excitatory and inhibitory) in order to study the time evolution of just one pool of neurons for each network node (Wong & Wang, 2006). It results that the dynamics of each network node described the temporal evolution of the opening probability of the NMDA channels.

ReducedWongWangExcInh

Reduced WongWang Exc-Inh (RWW_EI) is a biologically-inspired two-dimensional (i.e., two state-variables ('S_E','S_I')) neural mass model describing the dynamics of a cortical area consisting of local networks of excitatory (AMPA) and inhibitory (GABA-A) populations of spiking neurons interconnected via NMDA synapses. These neurons are organized into an inhibitory population accounting for 20% of the neurons and an excitatory population accounting for 80% of the neurons.

ReducedWongWangTvboptim

The Reduced Wong-Wang (RWW) model is a biophysically-based neural mass model that describes the dynamics of NMDA-mediated synaptic gating (Deco et al., 2013). It captures the slow dynamics relevant for resting-state fMRI and has been widely used for modeling whole-brain functional connectivity. RWW is the dynamical mean-field (DMF) reduction of the Wong & Wang (2006) spiking model. Each network node is described by a single state variable S representing the average NMDA synaptic gating (fraction of open channels). The total synaptic input x combines local excitatory recurrence (w), external input (I_o), and long-range network coupling (instant and delayed) scaled by the synaptic coupling strength J_N. The sigmoid transfer function H(x) maps the total input current to the average population firing rate. In a network setting, the global coupling strength G (a coupling parameter) scales the structural connectivity weights that modulate inter-regional communication.

RiddledBasins

Time-periodically forced four-dimensional system exhibiting riddled basins of attraction: arbitrarily small neighborhoods contain points converging to distinct attractors, illustrating extreme sensitivity in basin structure.

Rikitake

Rikitake two-disk dynamo model for geomagnetic polarity reversals. Exhibits chaotic switching between polarity states through nonlinear coupling of mechanical and electromagnetic components.

Roessler

This three-dimensional continuous system is due to Rössler [^Rössler1976]. It is a system that by design behaves similarly to the Lorenz system and displays a strange attractor. However, it is easier to analyze qualitatively, as for example the attractor is composed of a single manifold. Default values are the same as the original paper.

RulkovMap

Two-dimensional phenomenological neuron map capable of spiking and bursting regimes.

Sakarya

Four-wing strange attractor born via merging of bistable attractors; exhibits robust chaos and symmetry-related lobe structure transitions under parameter variation.

Shinriki

Shinriki oscillator (stiff chaotic electronic circuit) with exponential nonlinearity. All parameters except R1 fixed to canonical literature values.

SprottDissipativeConservative

Mixed dissipative/conservative three-dimensional system where initial condition choice yields either quasi-periodic torus motion or chaotic attractor behavior.

StandardMap

Chirikov standard map (area-preserving map on the 2-torus). At k≈0.971635 the golden invariant torus breaks. Equations are expressed with modulo 2π on both variables.

StefanescuJirsa2D

Reduced set of FitzHugh-Nagumo oscillators (Stefanescu & Jirsa 2008). A low-dimensional description of globally coupled heterogeneous excitatory and inhibitory FitzHugh-Nagumo populations. The reduction projects an infinite population onto `number_of_modes` modes and yields per-mode scalar state variables (xi, eta) for the excitatory and (alpha, beta) for the inhibitory subnetwork. The mode-coupling matrices Aik, Bik, Cik and the per-mode inputs IE_i, II_i, m_i, n_i, e_i, f_i are derived offline by Gaussian quadrature over the heterogeneity distribution N(mu, sigma^2); see TVB reference implementation `tvb.simulator.models.stefanescu_jirsa.ReducedSetFitzHughNagumo`.

StefanescuJirsa3D

Reduced set of Hindmarsh-Rose oscillators (Stefanescu & Jirsa 2008). A low-dimensional description of globally coupled heterogeneous excitatory and inhibitory Hindmarsh-Rose populations. The reduction projects an infinite population onto `number_of_modes` modes and yields per-mode scalar state variables (xi, eta, tau) for the excitatory and (alpha, beta, gamma) for the inhibitory subnetwork. The mode-coupling matrices A_ik, B_ik, C_ik and the per-mode coefficients a_i, b_i, c_i, d_i, e_i, f_i, h_i, p_i, IE_i, II_i, m_i, n_i are derived offline by Gaussian quadrature over the heterogeneity distribution N(mu, sigma^2); see TVB reference implementation `tvb.simulator.models.stefanescu_jirsa.ReducedSetHindmarshRose`.

StommelThermohaline

Two-box thermohaline circulation model capturing temperature (T) and salinity (S) driven density contrasts leading to multiple equilibria in overturning strength.

StuartLandauOscillator

Normal form of supercritical Hopf bifurcation in Cartesian coordinates; radius dynamics decouple giving stable limit cycle of radius sqrt(mu) for mu > 0.

SupHopf

Supercritical Hopf (SUPH) is an oscillatory/excitable neural mass model that describe the normal form of a supercritical Hopf bifurcation in Cartesian coordinates (Kuznetsov, 2013; Deco et al., 2017). This normal form has a supercritical bifurcation at 'a=0' with 'a' the bifurcation parameter in the model. So for 'a < 0', the local dynamics has a stable fixed point and the system corresponds to a damped oscillatory state, whereas for 'a > 0', the local dynamics enters in a stable limit cycle and the system switches to an oscillatory state.

SwingingAtwood

The swinging Atwood machine couples a pendulum to a vertically moving mass via a single inextensible string over a pulley. Competing exchanges between vertical and angular motion create a rich mixed phase space with coexisting regular islands and chaotic seas, providing a canonical laboratory example of low-dimensional Hamiltonian chaos and resonance overlap.

TentMap

Piecewise-linear tent map on [0,1]. At μ=2 it is conjugate to the logistic map with r=4.

ThomasCyclical

Thomas' cyclically symmetric attractor: a smooth three-dimensional system with rotational symmetry across cyclic permutations (x,y,z). Reduces to the labyrinth system when b = 0. Displays a strange attractor with intertwined scroll-like lobes.

TowelMap

Folded-towel hyperchaotic map with two positive Lyapunov exponents, introduced by Rössler. Named for the towel-like appearance of its attractor projections.

TsodyksMarkram

Tsodyks-Markram mean-field model of short-term synaptic plasticity. A three-variable neural mass model with firing rate E, synaptic depression variable x, and facilitation variable u. The transfer function uses a softplus nonlinearity α·log(1 + exp(s/α)). Reproduces the dynamics from BifurcationKit.jl/examples/TMModel.jl.

Ueda

Classic forced nonlinear oscillator (Ueda system) producing a broken-egg stroboscopic attractor under periodic sampling. Among early documented examples of deterministic chaos in driven oscillatory systems.

UlamRing4

Four unidirectionally coupled quadratic maps on a ring using the classic Ulam map f(x)=2 - x^2 as local dynamics; higher N can be constructed analogously.

VanDerPolForced

The forced van der Pol oscillator features nonlinear (amplitude dependent) damping and sinusoidal forcing. The unforced system has a unique attracting relaxation cycle; with forcing it displays entrainment, quasi-periodicity or chaos depending on parameters.

WilsonCowan

The Wilson and Cowan model consists of two populations or masses, one excitatory and one inhibitory, that are described by their mean firings rates E and I respectively (Wilson and Cowan, 1972, 1973). This model is the minimal representation of a NMM with a coarse-grained description of the overall activity of a large-scale neuronal network, as opposed to being a detailed biophysical model. While employing just two differential equations, it has been used to build various biophysically realistic models (Liley et al., 1999; Daffertshofer and van Wijk, 2011). Key parameters in the model are the strength of connectivity between each subtype of population (excitatory and inhibitory) and the strength of input to each subpopulation. The Input parameters P and Q also provide the entry point for local and long-range connectivity, that is, the activity coming from neighboring and distant populations respectively. Varying Input and connectivity generates a diversity of dynamical behaviors that are representative of observed activity in the brain, like multistability, oscillations, traveling waves and spatial patterns. We consider the transmission parameters of the excitatory population to be glutamatergic and therefore to be modified by glutamatergic receptors. The inhibitory population is considered as GABAergic. Note: - Equations and parameter names are taken from (Wilson and Cowan, 1972 and Sanz-Leon et al., 2015) - Default parameters are taken from Fig. 4 p.10 (Wilson and Cowan, 1972) - The model in Sanz-Leon et., 2015 includes more parameters than the original model, which can be traced in the description of the parameters.

ZaslavskiiMap

Two-dimensional dissipative map exhibiting chaotic attractors for standard parameter sets; sometimes mischaracterized as the “simplest” strange attractor.

ZerlautAdaptationFirstOrder

No description.

ZerlautAdaptationSecondOrder

Second-order mean-field model of a conductance-based network of adaptive exponential integrate-and-fire (AdEx) excitatory and inhibitory neurons (di Volo et al. 2018, Zerlaut et al. 2018). It tracks the mean firing rates of the excitatory (E) and inhibitory (I) populations, their second-order statistics (the covariance matrix C_ee, C_ei, C_ii), the adaptation currents (W_e, W_i) and an Ornstein-Uhlenbeck noise drift (ou_drift). The population firing rate is given by a semi-analytic transfer function (TF) whose argument is the membrane-potential fluctuation regime (mu_V, sigma_V, T_V) and a phenomenological voltage threshold fitted by a second-order polynomial. The first and second derivatives of the transfer function with respect to the pre-synaptic rates are evaluated with the SAME central finite-difference stencil (df = 1e-7) used in the original TVB implementation, so the drift is reproduced bit-for-bit.

ZetterbergJansen

No description.

hhcell_1

No description.

AdaptiveExponentialIF

Adaptive Exponential Integrate-and-Fire model from NeuroML2 adExIaFCell. Brette R and Gerstner W (2005) J Neurophysiol 94:3637-3642.

Two-Population Reduced Wong-Wang Model

Biophysically-based neural mass model with explicit excitatory (E) and inhibitory (I) populations. Each population has separate synaptic gating variables (S_e, S_i) and transfer functions. Enables independent control of E-I balance via the J_i parameter.

Models

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