generate

Functions:

Name	Description
chua	Vector field of Chua’s circuit, with the piecewise-linear diode “h(x) = m1 x + (m0 - m1) (
ctrnn	Vector field of a continuous-time recurrent neural network dx/dt = -x + g J tanh(x).
double_pendulum	Vector field of the double pendulum.
duffing	Vector field of the forced Duffing oscillator.
euler_spiral	Vector field of the Euler spiral (clothoid).
fitzhugh_nagumo	Vector field of the FitzHugh-Nagumo neuron model.
hindmarsh_rose	Vector field of the Hindmarsh-Rose neuron model.
kuramoto	Vector field of the Kuramoto model dθ_i/dt = ω_i + Σ_j K_ij sin(θ_j - θ_i).
lorenz	Vector field of the Lorenz system.
lorenz96	Vector field of the Lorenz 96 system.
lotka_volterra	Vector field of the generalized Lotka-Volterra system dx/dt = x (sigma - rho x) + S.
mackey_glass	Generate a Mackey-Glass chaotic time series via forward Euler integration.
morris_lecar	Vector field of the Morris-Lecar neuron model.
pyloric	Vector field of the three-neuron pyloric circuit of the crustacean stomatogastric ganglion (Prinz, Bucher & Marder 2004), in ms and mV.
rk45	Integrate dx/dt = f(t, x) via the adaptive Dormand-Prince RK45 method.
rossler	Vector field of the Rössler system.
rulkov_map	Generate a time series of globally coupled Rulkov map neurons.
three_scroll	Vector field of Li’s three-scroll system.
to_xy	Convert double pendulum angles to Cartesian coordinates.

chua

chua(alpha: float, beta: float, m0: float, m1: float)

Vector field of Chua’s circuit, with the piecewise-linear diode h(x) = m1 x + (m0 - m1) (|x + 1| - |x - 1|) / 2.

Parameters:

Name	Type	Description	Default
alpha	float	Ratio of capacitances (typical: 15.6).	required
beta	float	Ratio of inductance to resistance (typical: 28).	required
m0	float	Diode slope for “	x
m1	float	Diode slope for “	x

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (x, y, z) of shape (3,), or (B, 3) for a batch.

ctrnn

ctrnn(g: float, J: np.ndarray)

Vector field of a continuous-time recurrent neural network dx/dt = -x + g J tanh(x).

Parameters:

Name	Type	Description	Default
g	float	Gain (chaos for g > 1 when J is large and random with variance 1 / N).	required
J	ndarray	Connectivity of shape (N, N); J[i, j] is the weight with which unit j drives unit i.	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state of shape (N,), or (B, N) for a batch.

double_pendulum

double_pendulum(m1: float, m2: float, L1: float, L2: float, g: float)

Vector field of the double pendulum.

Parameters:

Name	Type	Description	Default
m1	float	Mass of first pendulum.	required
m2	float	Mass of second pendulum.	required
L1	float	Length of first pendulum.	required
L2	float	Length of second pendulum.	required
g	float	Gravitational acceleration.	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (theta1, theta2, omega1, omega2) of shape (4,), or (B, 4) for a batch.

duffing

duffing(alpha: float, beta: float, delta: float, gamma: float, omega: float)

Vector field of the forced Duffing oscillator.

Parameters:

Name	Type	Description	Default
alpha	float	Linear stiffness (typical: -1).	required
beta	float	Cubic stiffness (typical: 1).	required
delta	float	Damping coefficient (typical: 0.3).	required
gamma	float	Forcing amplitude (typical: 0.5 for chaos).	required
omega	float	Forcing angular frequency (typical: 1.2).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (x, v) of shape (2,), or (B, 2) for a batch.

euler_spiral

euler_spiral(a: float)

Vector field of the Euler spiral (clothoid).

Parameters:

Name	Type	Description	Default
a	float	Rate at which the curvature grows (typical: 1).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (x, y, theta) of shape (3,), or (B, 3) for a batch.

fitzhugh_nagumo

fitzhugh_nagumo(a: float, b: float, epsilon: float, I: float)

Vector field of the FitzHugh-Nagumo neuron model.

Parameters:

Name	Type	Description	Default
a	float	Offset of the recovery variable (typical: 0.7).	required
b	float	Damping of the recovery variable (typical: 0.8).	required
epsilon	float	Time scale of the recovery variable (typical: 0.08).	required
I	float	Applied current (typical: 0.5 for spiking).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (v, w) of shape (2,), or (B, 2) for a batch.

hindmarsh_rose

hindmarsh_rose(a: float, b: float, c: float, d: float, r: float, s: float, x_R: float, I: float)

Vector field of the Hindmarsh-Rose neuron model.

Parameters:

Name	Type	Description	Default
a	float	Cubic coefficient of the membrane potential (typical: 1).	required
b	float	Quadratic coefficient of the membrane potential (typical: 3).	required
c	float	Constant drive of the recovery variable (typical: 1).	required
d	float	Quadratic coupling of the potential into the recovery variable (typical: 5).	required
r	float	Time scale of the slow adaptation current (typical: 0.006).	required
s	float	Adaptation strength (typical: 4).	required
x_R	float	Resting potential (typical: -1.6).	required
I	float	Applied current (typical: 3.25 for chaotic bursting).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (x, y, z) of shape (3,), or (B, 3) for a batch: membrane potential, recovery variable and adaptation current.

kuramoto

kuramoto(omega: np.ndarray, K: np.ndarray)

Vector field of the Kuramoto model dθ_i/dt = ω_i + Σ_j K_ij sin(θ_j - θ_i).

Parameters:

Name	Type	Description	Default
omega	ndarray	Natural frequencies of shape (N,).	required
K	ndarray	Coupling of shape (N, N); K[i, j] is the strength with which oscillator j drives oscillator i (typical: k / N everywhere).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for unwrapped phases of shape (N,), or (B, N) for a batch.

lorenz

lorenz(sigma: float, rho: float, beta: float)

Vector field of the Lorenz system.

Parameters:

Name	Type	Description	Default
sigma	float	Prandtl number (typical: 10).	required
rho	float	Rayleigh number (typical: 28).	required
beta	float	Geometric factor (typical: 8/3).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (x, y, z) of shape (3,), or (B, 3) for a batch.

lorenz96

lorenz96(F: float)

Vector field of the Lorenz 96 system.

Parameters:

Name	Type	Description	Default
F	float	Forcing constant (typical: 8 for chaos).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state of shape (D,), or (B, D) for a batch, with D >= 4.

lotka_volterra

lotka_volterra(rho: np.ndarray, sigma: float | np.ndarray, S: float | np.ndarray)

Vector field of the generalized Lotka-Volterra system dx/dt = x (sigma - rho x) + S.

Parameters:

Name	Type	Description	Default
rho	ndarray	Interaction matrix of shape (N, N); rho[i, j] is the effect of species j on species i.	required
sigma	float | ndarray	Growth rate, scalar or of shape (N,) (typical: 1).	required
S	float | ndarray	Constant input, scalar or of shape (N,) (typical: 0; small positive values keep winnerless competition switching).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state of shape (N,), or (B, N) for a batch.

mackey_glass

mackey_glass(tau: float, n: int, beta: float, gamma: float, x0: float, dt: float, t_max: int)

Generate a Mackey-Glass chaotic time series via forward Euler integration.

Parameters:

Name	Type	Description	Default
tau	float	Delay parameter (typical: 17 for chaos).	required
n	int	Nonlinearity exponent (typical: 10).	required
beta	float	Feedback strength (typical: 0.2).	required
gamma	float	Decay rate (typical: 0.1).	required
x0	float	Initial condition.	required
dt	float	Integration time step.	required
t_max	int	Maximum time.	required

Returns:

Name	Type	Description
t	ndarray	Time array.
x	ndarray	1D time series.

morris_lecar

morris_lecar(C: float, g_L: float, g_Ca: float, g_K: float, V_L: float, V_Ca: float, V_K: float, V1: float, V2: float, V3: float, V4: float, phi: float, I: float)

Vector field of the Morris-Lecar neuron model.

Parameters:

Name	Type	Description	Default
C	float	Membrane capacitance (typical: 20).	required
g_L	float	Leak conductance (typical: 2).	required
g_Ca	float	Calcium conductance (typical: 4.4).	required
g_K	float	Potassium conductance (typical: 8).	required
V_L	float	Leak reversal potential (typical: -60).	required
V_Ca	float	Calcium reversal potential (typical: 120).	required
V_K	float	Potassium reversal potential (typical: -84).	required
V1	float	Half-activation voltage of the calcium channel (typical: -1.2).	required
V2	float	Slope of the calcium activation (typical: 18).	required
V3	float	Half-activation voltage of the potassium channel (typical: 2).	required
V4	float	Slope of the potassium activation (typical: 30).	required
phi	float	Rate of the potassium activation (typical: 0.04).	required
I	float	Applied current (typical: 100 for spiking).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (V, n) of shape (2,), or (B, 2) for a batch: membrane potential and potassium activation.

pyloric

pyloric(g_mem: np.ndarray, g_glu: np.ndarray, g_ach: np.ndarray)

Vector field of the three-neuron pyloric circuit of the crustacean stomatogastric ganglion (Prinz, Bucher & Marder 2004), in ms and mV.

Parameters:

Name	Type	Description	Default
g_mem	ndarray	Maximal conductances in mS/cm^2 of shape (3, 8): rows are AB/PD, LP, PY and columns are Na, CaT, CaS, A, KCa, Kd, H, leak (typical: [[400, 2.5, 6, 50, 10, 100, 0.01, 0], [100, 0, 8, 40, 5, 75, 0.05, 0.02], [100, 2.5, 2, 50, 0, 125, 0.05, 0.01]]).	required
g_glu	ndarray	Fast glutamatergic synapse strengths in nS of shape (3, 3); g_glu[i, j] is the synapse from neuron j onto neuron i (typical: g_glu[1, 0] = 30 (AB to LP), g_glu[2, 0] = 3 (AB to PY), g_glu[0, 1] = 10 (LP to AB/PD), g_glu[2, 1] = 3 (LP to PY), g_glu[1, 2] = 3 (PY to LP), the rest 0, for a PD, LP, PY rhythm of period about 1.7 s).	required
g_ach	ndarray	Slow cholinergic synapse strengths in nS of shape (3, 3), indexed like g_glu (typical: g_ach[1, 0] = 3 (PD to LP), g_ach[2, 0] = 30 (PD to PY), the rest 0).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state of shape (45,), or (B, 45) for a batch; x.reshape(15, 3) has one column per neuron and rows: membrane potential, intracellular [Ca] in uM, the gates m and h of Na, m and h of CaT, m and h of CaS, m and h of A, m of KCa, n of Kd, m of H, and the glutamatergic and cholinergic synaptic activations of the neuron as presynaptic cell (typical initial state: V = -50, [Ca] = 0.05, all gates and synaptic activations 0, discard the first 3 s as transient).

rk45

rk45(f: Callable[[float, np.ndarray], np.ndarray], X0: np.ndarray, dt: float, t_max: float, rtol: float = 1e-06, atol: float = 1e-09)

Integrate dx/dt = f(t, x) via the adaptive Dormand-Prince RK45 method.

Steps are chosen by the error alone, and the uniform grid of spacing dt is filled in by the method’s dense output. A batch of initial conditions is integrated together, with steps that keep every member within the tolerance.

Parameters:

Name	Type	Description	Default
f	Callable[[float, ndarray], ndarray]	Right-hand side of the system, called with states of the shape of X0.	required
X0	ndarray	Initial condition of shape (D,), or (B, D) for a batch.	required
dt	float	Sampling time step.	required
t_max	float	End time, excluded from the grid.	required
rtol	float	Relative tolerance of the local error.	1e-06
atol	float	Absolute tolerance of the local error.	1e-09

Returns:

Name	Type	Description
t	ndarray	Time array of shape (N,).
X	ndarray	Trajectory of shape (N, D), or (B, N, D) for a batch.

Raises:

Type	Description
ValueError	If X0 is not of shape (D,) or (B, D).

rossler

rossler(a: float, b: float, c: float)

Vector field of the Rössler system.

Parameters:

Name	Type	Description	Default
a	float	Spiral growth rate (typical: 0.2).	required
b	float	Constant injection into z (typical: 0.2).	required
c	float	Folding threshold (typical: 5.7).	required

Returns:

Name	Type	Description
f	Callable[[float, ndarray], ndarray]	Right-hand side f(t, x) for a state (x, y, z) of shape (3,), or (B, 3) for a batch.

rulkov_map

rulkov_map(alpha: float, mu: float, sigma: float, epsilon: float, X0: np.ndarray, n_max: int)

Generate a time series of globally coupled Rulkov map neurons.

Parameters:

Name	Type	Description	Default
alpha	float	Nonlinearity of the fast variable (typical: 4.1 for chaotic bursting).	required
mu	float	Time scale of the slow variable (typical: 0.001).	required
sigma	float	Drive of the slow variable (typical: -1.5).	required
epsilon	float	Coupling through the mean of the fast variables (typical: 0).	required
X0	ndarray	Initial condition of shape (N, 2), one (x, y) per unit.	required
n_max	int	Number of steps.	required

Returns:

Name	Type	Description
n	ndarray	Step array.
X	ndarray	Time series of shape (n_max, N, 2).

three_scroll

three_scroll(a: float, k: float, c: float, d: float, e: float, f: float)

Vector field of Li’s three-scroll system.

Parameters:

Name	Type	Description	Default
a	float	Coupling of x towards y (typical: 40).	required
k	float	Gain of x in dy/dt (typical: 55).	required
c	float	Gain of z in dz/dt (typical: 1.9 for chaos; Li’s 11/6 gives a limit cycle).	required
d	float	Coefficient of x z in dx/dt (typical: 0.16).	required
e	float	Coefficient of x^2 in dz/dt (typical: 0.65).	required
f	float	Gain of y in dy/dt (typical: 20).	required

Returns:

Name	Type	Description
rhs	Callable[[float, ndarray], ndarray]	Right-hand side rhs(t, x) for a state (x, y, z) of shape (3,), or (B, 3) for a batch.

to_xy

to_xy(L1: float, L2: float, theta1: np.ndarray, theta2: np.ndarray)

Convert double pendulum angles to Cartesian coordinates.

Parameters:

Name	Type	Description	Default
L1	float	Length of first pendulum.	required
L2	float	Length of second pendulum.	required
theta1	ndarray	Angle of first pendulum.	required
theta2	ndarray	Angle of second pendulum.	required

Returns:

Name	Type	Description
x1	ndarray	x-coordinate of first pendulum.
y1	ndarray	y-coordinate of first pendulum.
x2	ndarray	x-coordinate of second pendulum.
y2	ndarray	y-coordinate of second pendulum.