models Package

A collection of neuronal dynamics models.

Specific models inherit from the abstract class Model, which in turn inherits from the class Trait from the tvb.basic.traits module.

base

This module defines the common imports and abstract base class for model definitions.

tvb.simulator.models.base.Model

Defines the abstract base class for neuronal models.

traits on this class:

tvb.simulator.models.base.ModelNumbaDfun

Base model for Numba-implemented dfuns. traits on this class:

epileptor

Hindmarsh-Rose-Jirsa Epileptor model.

tvb.simulator.models.epileptor.Epileptor

The Epileptor is a composite neural mass model of six dimensions which has been crafted to model the phenomenology of epileptic seizures. (see [Jirsaetal_2014])

Equations and default parameters are taken from [Jirsaetal_2014].

Table 1
Parameter	Value
I_rest1	3.1
I_rest2	0.45
r	0.00035
x_0	-1.6
slope	0.0
Integration parameter
dt	0.1
simulation_length	4000
Noise
nsig	[0., 0., 0., 1e-3, 1e-3, 0.]
Jirsa et al. 2014

[Jirsaetal_2014]

(1, 2) Jirsa, V. K.; Stacey, W. C.; Quilichini, P. P.; Ivanov, A. I.; Bernard, C. On the nature of seizure dynamics. Brain, 2014.

Epileptor.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

Variables of interest to be used by monitors: -y[0] + y[3]

\[\begin{split}\dot{x_{1}} &=& y_{1} - f_{1}(x_{1}, x_{2}) - z + I_{ext1} \\ \dot{y_{1}} &=& c - d x_{1}^{2} - y{1} \\ \dot{z} &=& \begin{cases} r(4 (x_{1} - x_{0}) - z-0.1 z^{7}) & \text{if } x<0 \\ r(4 (x_{1} - x_{0}) - z) & \text{if } x \geq 0 \end{cases} \\ \dot{x_{2}} &=& -y_{2} + x_{2} - x_{2}^{3} + I_{ext2} + 0.002 g - 0.3 (z-3.5) \\ \dot{y_{2}} &=& 1 / \tau (-y_{2} + f_{2}(x_{2}))\\ \dot{g} &=& -0.01 (g - 0.1 x_{1})\end{split}\]

where:

\[\begin{split}f_{1}(x_{1}, x_{2}) = \begin{cases} a x_{1}^{3} - b x_{1}^2 & \text{if } x_{1} <0\\ -(slope - x_{2} + 0.6(z-4)^2) x_{1} &\text{if }x_{1} \geq 0 \end{cases}\end{split}\]

and:

\[\begin{split}f_{2}(x_{2}) = \begin{cases} 0 & \text{if } x_{2} <-0.25\\ a_{2}(x_{2} + 0.25) & \text{if } x_{2} \geq -0.25 \end{cases}\end{split}\]

Note Feb. 2017: the slow permittivity variable can be modify to account for the time difference between interictal and ictal states (see [Proixetal_2014]).

[Proixetal_2014]

Proix, T.; Bartolomei, F; Chauvel, P; Bernard, C; Jirsa, V.K. * Permittivity coupling across brain regions determines seizure recruitment in partial epilepsy.* J Neurosci 2014, 34:15009-21.

traits on this class:

Iext (Iext)

External input current to the first population

default: [ 3.1]

range: low = 1.5 ; high = 5.0

Iext2 (Iext2)

External input current to the second population

default: [ 0.45]

range: low = 0.0 ; high = 1.0

Kf (K_f)

Correspond to the coupling scaling on a fast time scale.

default: [ 0.]

range: low = 0.0 ; high = 4.0

Ks (K_s)

Permittivity coupling, that is from the fast time scale toward the slow time scale

default: [ 0.]

range: low = -4.0 ; high = 4.0

Kvf (K_vf)

Coupling scaling on a very fast time scale.

default: [ 0.]

range: low = 0.0 ; high = 4.0

a (a)

Coefficient of the cubic term in the first state variable

default: [1]

aa (aa)

Linear coefficient in fifth state variable

default: [6]

b (b)

Coefficient of the squared term in the first state variabel

default: [3]

bb (bb)

Linear coefficient of lowpass excitatory coupling in fourth state variable

default: [2]

c (c)

Additive coefficient for the second state variable, called \(y_{0}\) in Jirsa paper

default: [1]

d (d)

Coefficient of the squared term in the second state variable

default: [5]

modification (modification)

When modification is True, then use nonlinear influence on z. The default value is False, i.e., linear influence.

default: [0]

r (r)

Temporal scaling in the third state variable, called \(1/\tau_{0}\) in Jirsa paper

default: [ 0.00035]

range: low = 0.0 ; high = 0.001

s (s)

Linear coefficient in the third state variable

default: [4]

slope (slope)

Linear coefficient in the first state variable

default: [ 0.]

range: low = -16.0 ; high = 6.0

state_variable_range (State variable ranges [lo, hi])

Typical bounds on state variables in the Epileptor model.

default: {‘y2’: array([ 0., 2.]), ‘g’: array([-1., 1.]), ‘z’: array([ 2., 5.]), ‘y1’: array([-20., 2.]), ‘x2’: array([-2., 0.]), ‘x1’: array([-2., 1.])}

tau (tau)

Temporal scaling coefficient in fifth state variable

default: [10]

tt (tt)

Time scaling of the whole system

default: [ 1.]

range: low = 0.001 ; high = 10.0

variables_of_interest (Variables watched by Monitors)

Quantities of the Epileptor available to monitor.

default: [‘x2 - x1’, ‘z’]

x0 (x0)

Epileptogenicity parameter

default: [-1.6]

range: low = -3.0 ; high = -1.0

tvb.simulator.models.epileptor.Epileptor2D

Two-dimensional reduction of the Epileptor.

Taking advantage of time scale separation and focusing on the slower time scale, the five-dimensional Epileptor reduces to a two-dimensional system (see [Proixetal_2014, Proixetal_2017]).

Note: the slow permittivity variable can be modify to account for the time difference between interictal and ictal states (see [Proixetal_2014]).

Equations and default parameters are taken from [Proixetal_2014]:

\[\begin{split}\dot{x_{1,i}} &=& - x_{1,i}^{3} - 2x_{1,i}^{2} + 1 - z_{i} + I_{ext1,i} \\ \dot{z_{i}} &=& r(h - z_{i})\end{split}\]

with

h = x_{0} + 3 / (exp((x_{1} + 0.5)/0.1)) if modification h = 4 (x_{1,i} - x_{0})

References:

[Proixetal_2014] Proix, T.; Bartolomei, F; Chauvel, P; Bernard, C; Jirsa, V.K. * Permittivity coupling across brain regions determines seizure recruitment in partial epilepsy.* J Neurosci 2014, 34:15009-21.

[Proixetal_2017] Proix, T.; Bartolomei, F; Guye, M.; Jirsa, V.K. Individual brain structure and modelling predict seizure propagation. Brain 2017, 140; 641–654.

traits on this class:

Iext (Iext)

External input current to the first state-variable.

default: [ 3.1]

range: low = 1.5 ; high = 5.0

Ks (K_s)

Permittivity coupling, that is from the fast time scale toward the slow time scale.

default: [ 0.]

range: low = -4.0 ; high = 4.0

Kvf (K_vf)

Coupling scaling on a very fast time scale.

default: [ 0.]

range: low = 0.0 ; high = 4.0

a (a)

Coefficient of the cubic term in the first state-variable.

default: [1]

b (b)

Coefficient of the squared term in the first state-variable.

default: [3]

c (c)

Additive coefficient for the second state-variable x_{2}, called \(y_{0}\) in Jirsa paper.

default: [1]

d (d)

Coefficient of the squared term in the second state-variable x_{2}.

default: [5]

modification (modification)

When modification is True, then use nonlinear influence on z. The default value is False, i.e., linear influence.

default: [0]

r (r)

Temporal scaling in the slow state-variable, called \(1\tau_{0}\) in Jirsa paper (see class Epileptor).

default: [ 0.00035]

range: low = 0.0 ; high = 0.001

slope (slope)

Linear coefficient in the first state-variable.

default: [ 0.]

range: low = -16.0 ; high = 6.0

state_variable_range (State variable ranges [lo, hi])

Typical bounds on state-variables in the Epileptor 2D model.

default: {‘x1’: array([-2., 1.]), ‘z’: array([ 2., 5.])}

tt (tt)

Time scaling of the whole system to the system in real time.

default: [ 1.]

range: low = 0.001 ; high = 1.0

variables_of_interest (Variables watched by Monitors)

Quantities of the Epileptor 2D available to monitor.

default: [‘x1’]

x0 (x0)

Epileptogenicity parameter.

default: [-1.6]

range: low = -3.0 ; high = -1.0

hopfield

Hopfield model with modifications following Golos & Daucé.

tvb.simulator.models.hopfield.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].

References:

[Hopfield1982]

Hopfield, J. J., Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. (USA) 79, 2554-2558, 1982.

[Hopfield1984]

Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-sate neurons, Proc. Nat. Acad. Sci. (USA) 81, 3088-3092, 1984.

See also, http://www.scholarpedia.org/article/Hopfield_network

Hopfield.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

Hopfield.configure()

Set the threshold as a state variable for a dynamical threshold.

Dynamic equations:

dfun equation

\[\begin{split}\dot{x_{i}} &= 1 / \tau_{x} (-x_{i} + c_0)\end{split}\]

dfun dynamic equation

\[\begin{split}\dot{x_{i}} &= 1 / \tau_{x} (-x_{i} + c_0(i)) \\ \dot{\\theta_{i}} &= 1 / \tau_{\theta_{i}} (-\theta + c_1(i))\end{split}\]

The phase-plane for the Hopfield model.

traits on this class:

dynamic (Dynamic)

Boolean value for static/dynamic threshold theta for (0/1).

default: [0]

range: low = 0 ; high = 1.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘x’: array([-1., 2.]), ‘theta’: array([ 0., 1.])}

tauT (\(\tau_{\theta}\))

The slow time-scale for threshold calculus :math:` heta`, state-variable of the model.

default: [ 5.]

range: low = 0.01 ; high = 100.0

taux (\(\tau_{x}\))

The fast time-scale for potential calculus \(x\), state-variable of the model.

default: [ 1.]

range: low = 0.01 ; high = 100.0

variables_of_interest (Variables watched by Monitors)

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random initial conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: [‘x’]

jansen_rit

Jansen-Rit and derivative models.

tvb.simulator.models.jansen_rit.JansenRit

The Jansen and Rit is a biologically inspired mathematical framework originally conceived to simulate the spontaneous electrical activity of neuronal assemblies, with a particular focus on alpha activity, for instance, as measured by EEG. Later on, it was discovered that in addition to alpha activity, this model was also able to simulate evoked potentials.

[JR_1995]

Jansen, B., H. and Rit V., G., Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns, Biological Cybernetics (73) 357:366, 1995.

[J_1993]

Jansen, B., Zouridakis, G. and Brandt, M., A neurophysiologically-based mathematical model of flash visual evoked potentials

The (\(y_4\), \(y_5\)) phase-plane for the Jansen and Rit model.

JansenRit.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

The dynamic equations were taken from [JR_1995]

\[\begin{split}\dot{y_0} &= y_3 \\ \dot{y_3} &= A a\,S[y_1 - y_2] - 2a\,y_3 - 2a^2\, y_0 \\ \dot{y_1} &= y_4\\ \dot{y_4} &= A a \,[p(t) + \alpha_2 J + S[\alpha_1 J\,y_0]+ c_0] -2a\,y - a^2\,y_1 \\ \dot{y_2} &= y_5 \\ \dot{y_5} &= B b (\alpha_4 J\, S[\alpha_3 J \,y_0]) - 2 b\, y_5 - b^2\,y_2 \\ S[v] &= \frac{2\, \nu_{max}}{1 + \exp^{r(v_0 - v)}}\end{split}\]

traits on this class:

A (A)

Maximum amplitude of EPSP [mV]. Also called average synaptic gain.

default: [ 3.25]

range: low = 2.6 ; high = 9.75

B (B)

Maximum amplitude of IPSP [mV]. Also called average synaptic gain.

default: [ 22.]

range: low = 17.6 ; high = 110.0

J (\(J\))

Average number of synapses between populations.

default: [ 135.]

range: low = 65.0 ; high = 1350.0

a (\(a\))

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.1]

range: low = 0.05 ; high = 0.15

a_1 (\(\alpha_1\))

Average probability of synaptic contacts in the feedback excitatory loop.

default: [ 1.]

range: low = 0.5 ; high = 1.5

a_2 (\(\alpha_2\))

Average probability of synaptic contacts in the slow feedback excitatory loop.

default: [ 0.8]

range: low = 0.4 ; high = 1.2

a_3 (\(\alpha_3\))

Average probability of synaptic contacts in the feedback inhibitory loop.

default: [ 0.25]

range: low = 0.125 ; high = 0.375

a_4 (\(\alpha_4\))

Average probability of synaptic contacts in the slow feedback inhibitory loop.

default: [ 0.25]

range: low = 0.125 ; high = 0.375

b (\(b\))

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.05]

range: low = 0.025 ; high = 0.075

mu (\(\mu_{max}\))

Mean input firing rate

default: [ 0.22]

range: low = 0.0 ; high = 0.22

nu_max (\(\nu_{max}\))

Determines the maximum firing rate of the neural population [s^-1].

default: [ 0.0025]

range: low = 0.00125 ; high = 0.00375

p_max (\(p_{max}\))

Maximum input firing rate.

default: [ 0.32]

range: low = 0.0 ; high = 0.32

p_min (\(p_{min}\))

Minimum input firing rate.

default: [ 0.12]

range: low = 0.0 ; high = 0.12

r (\(r\))

Steepness of the sigmoidal transformation [mV^-1].

default: [ 0.56]

range: low = 0.28 ; high = 0.84

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘y1’: array([-500., 500.]), ‘y0’: array([-1., 1.]), ‘y3’: array([-6., 6.]), ‘y2’: array([-50., 50.]), ‘y5’: array([-500., 500.]), ‘y4’: array([-20., 20.])}

v0 (\(v_0\))

Firing threshold (PSP) for which a 50% firing rate is achieved. In other words, it is the value of the average membrane potential corresponding to the inflection point of the sigmoid [mV]. The usual value for this parameter is 6.0.

default: [ 5.52]

range: low = 3.12 ; high = 6.0

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(y0 = 0\), \(y1 = 1\), \(y2 = 2\), \(y3 = 3\), \(y4 = 4\), and \(y5 = 5\)

default: [‘y0’, ‘y1’, ‘y2’, ‘y3’]

tvb.simulator.models.jansen_rit.ZetterbergJansen

Zetterberg et al derived a model inspired by the Wilson-Cowan equations. It served as a basis for the later, better known Jansen-Rit model.

[ZL_1978]

Zetterberg LH, Kristiansson L and Mossberg K. Performance of a Model for a Local Neuron Population. Biological Cybernetics 31, 15-26, 1978.

[JB_1995]

Jansen, B., H. and Rit V., G., Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns, Biological Cybernetics (73) 357:366, 1995.

[JB_1993]

Jansen, B., Zouridakis, G. and Brandt, M., A neurophysiologically-based mathematical model of flash visual evoked potentials

[M_2007]

Moran

[S_2010]

Spiegler

[A_2012]

Auburn

traits on this class:

He (\(H_e\))

Maximum amplitude of EPSP [mV]. Also called average synaptic gain.

default: [ 3.25]

range: low = 2.6 ; high = 9.75

Hi (\(H_i\))

Maximum amplitude of IPSP [mV]. Also called average synaptic gain.

default: [ 22.]

range: low = 17.6 ; high = 110.0

P (\(P\))

Maximum firing rate to the pyramidal population [ms^-1]. (External stimulus. Constant intensity.Entry point for coupling.)

default: [ 0.12]

range: low = 0.0 ; high = 0.35

Q (\(Q\))

Maximum firing rate to the interneurons population [ms^-1]. (External stimulus. Constant intensity.Entry point for coupling.)

default: [ 0.12]

range: low = 0.0 ; high = 0.35

U (\(U\))

Maximum firing rate to the stellate population [ms^-1]. (External stimulus. Constant intensity.Entry point for coupling.)

default: [ 0.12]

range: low = 0.0 ; high = 0.35

e0 (\(e_0\))

Half of the maximum population mean firing rate [ms^-1].

default: [ 0.0025]

range: low = 0.00125 ; high = 0.00375

gamma_1 (\(\gamma_1\))

Average number of synapses between populations (pyramidal to stellate).

default: [ 135.]

range: low = 65.0 ; high = 1350.0

gamma_1T (\(\gamma_{1T}\))

Coupling factor from the extrinisic input to the spiny stellate population.

default: [ 1.]

range: low = 0.0 ; high = 1000.0

gamma_2 (\(\gamma_2\))

Average number of synapses between populations (stellate to pyramidal).

default: [ 108.]

range: low = 0.0 ; high = 200

gamma_2T (\(\gamma_{2T}\))

Coupling factor from the extrinisic input to the pyramidal population.

default: [ 1.]

range: low = 0.0 ; high = 1000.0

gamma_3 (\(\gamma_3\))

Connectivity constant (pyramidal to interneurons)

default: [ 33.75]

range: low = 0.0 ; high = 200

gamma_3T (\(\gamma_{3T}\))

Coupling factor from the extrinisic input to the inhibitory population.

default: [ 1.]

range: low = 0.0 ; high = 1000.0

gamma_4 (\(\gamma_4\))

Connectivity constant (interneurons to pyramidal)

default: [ 33.75]

range: low = 0.0 ; high = 200

gamma_5 (\(\gamma_5\))

Connectivity constant (interneurons to interneurons)

default: [15]

range: low = 0.0 ; high = 100

ke (\(\kappa_e\))

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.1]

range: low = 0.05 ; high = 0.15

ki (\(\kappa_i\))

Reciprocal of the time constant of passive membrane and all other spatially distributed delays in the dendritic network [ms^-1]. Also called average synaptic time constant.

default: [ 0.05]

range: low = 0.025 ; high = 0.075

rho_1 (\(\rho_1\))

Steepness of the sigmoidal transformation [mV^-1].

default: [ 0.56]

range: low = 0.28 ; high = 0.84

rho_2 (\(\rho_2\))

Firing threshold (PSP) for which a 50% firing rate is achieved. In other words, it is the value of the average membrane potential corresponding to the inflection point of the sigmoid [mV]. Population mean firing threshold.

default: [ 6.]

range: low = 3.12 ; high = 10.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘v1’: array([-100., 100.]), ‘v2’: array([-100., 50.]), ‘v3’: array([-100., 6.]), ‘v4’: array([-100., 20.]), ‘v5’: array([-100., 20.]), ‘v6’: array([-100., 20.]), ‘v7’: array([-100., 20.]), ‘y1’: array([-500., 500.]), ‘y3’: array([-100., 6.]), ‘y2’: array([-100., 6.]), ‘y5’: array([-500., 500.]), ‘y4’: array([-100., 20.])}

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(v_6 = 0\), \(v_7 = 1\), \(v_2 = 2\), \(v_3 = 3\), \(v_4 = 4\), and \(v_5 = 5\)

default: [‘v6’, ‘v7’, ‘v2’, ‘v3’, ‘v4’, ‘v5’]

larter_breakspear

Larter-Breakspear model based on the Morris-Lecar equations.

tvb.simulator.models.larter_breakspear.LarterBreakspear

A modified Morris-Lecar model that includes a third equation which simulates the effect of a population of inhibitory interneurons synapsing on the pyramidal cells.

[Larteretal_1999]

Larter et.al. A coupled ordinary differential equation lattice model for the simulation of epileptic seizures. Chaos. 9(3): 795, 1999.

[Breaksetal_2003_a]

Breakspear, M.; Terry, J. R. & Friston, K. J. Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in an onlinear model of neuronal dynamics. Neurocomputing 52–54 (2003).151–158

[Breaksetal_2003_b]

(1, 2, 3) M. J. Breakspear et.al. Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics. Network: Computation in Neural Systems 14: 703-732, 2003.

[Honeyetal_2007]

Honey, C.; Kötter, R.; Breakspear, M. & Sporns, O. * Network structure of cerebral cortex shapes functional connectivity on multiple time scales*. (2007) PNAS, 104, 10240

[Honeyetal_2009]

Honey, C. J.; Sporns, O.; Cammoun, L.; Gigandet, X.; Thiran, J. P.; Meuli, R. & Hagmann, P. Predicting human resting-state functional connectivity from structural connectivity. (2009), PNAS, 106, 2035-2040

[Alstottetal_2009]

Alstott, J.; Breakspear, M.; Hagmann, P.; Cammoun, L. & Sporns, O. Modeling the impact of lesions in the human brain. (2009)), PLoS Comput Biol, 5, e1000408

Equations and default parameters are taken from [Breaksetal_2003_b]. All equations and parameters are non-dimensional and normalized. For values of d_v < 0.55, the dynamics of a single column settles onto a solitary fixed point attractor.

Parameters used for simulations in [Breaksetal_2003_a] Table 1. Page 153. Two nodes were coupled. C=0.1

Table 1
Parameter	Value
I	0.3
a_ee	0.4
a_ei	0.1
a_ie	1.0
a_ne	1.0
a_ni	0.4
r_NMDA	0.2
delta	0.001
Breakspear et al. 2003

Table 2
Parameter	Value
gK	2.0
gL	0.5
gNa	6.7
gCa	1.0
a_ne	1.0
a_ni	0.4
a_ee	0.36
a_ei	2.0
a_ie	2.0
VK	-0.7
VL	-0.5
VNa	0.53
VCa	1.0
phi	0.7
b	0.1
I	0.3
r_NMDA	0.25
C	0.1
TCa	-0.01
d_Ca	0.15
TK	0.0
d_K	0.3
VT	0.0
ZT	0.0
TNa	0.3
d_Na	0.15
d_V	0.65
d_Z	d_V
QV_max	1.0
QZ_max	1.0
Alstott et al. 2009

NOTES about parameters

\(\delta_V\) : for \(\delta_V\) < 0.55, in an uncoupled network, the system exhibits fixed point dynamics; for 0.55 < \(\delta_V\) < 0.59, limit cycle attractors; and for \(\delta_V\) > 0.59 chaotic attractors (eg, d_V=0.6,aee=0.5,aie=0.5, gNa=0, Iext=0.165)

\(\delta_Z\) this parameter might be spatialized: ones(N,1).*0.65 + modn*(rand(N,1)-0.5);

\(C\) The long-range coupling \(\delta_C\) is ‘weak’ in the sense that the model is well behaved for parameter values for which C < a_ee and C << a_ie.

The (\(V\), \(W\)) phase-plane for the Larter-Breakspear model.

LarterBreakspear.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

Dynamic equations:

\[\begin{split}\dot{V}_k & = - (g_{Ca} + (1 - C) \, r_{NMDA} \, a_{ee} \, Q_V + C \, r_{NMDA} \, a_{ee} \, \langle Q_V\rangle^{k}) \, m_{Ca} \, (V - VCa) \\ & \,\,- g_K \, W \, (V - VK) - g_L \, (V - VL) \\ & \,\,- (g_{Na} \, m_{Na} + (1 - C) \, a_{ee} \, Q_V + C \, a_{ee} \, \langle Q_V\rangle^{k}) \,(V - VNa) \\ & \,\,- a_{ie} \, Z \, Q_Z + a_{ne} \, I, \\ & \\ \dot{W}_k & = \phi \, \dfrac{m_K - W}{\tau_{K}},\\ & \nonumber\\ \dot{Z}_k &= b (a_{ni}\, I + a_{ei}\,V\,Q_V),\\ Q_{V} &= Q_{V_{max}} \, (1 + \tanh\left(\dfrac{V_{k} - VT}{\delta_{V}}\right)),\\ Q_{Z} &= Q_{Z_{max}} \, (1 + \tanh\left(\dfrac{Z_{k} - ZT}{\delta_{Z}}\right)),\end{split}\]\[See Equations (7), (3), (6) and (2) respectively in [Breaksetal_2003_a]_. Pag: 705-706\]

traits on this class:

C (\(C\))

Strength of excitatory coupling. Balance between internal and local (and global) coupling strength. C > 0 introduces interdependences between consecutive columns/nodes. C=1 corresponds to maximum coupling between node and no self-coupling. This strenght should be set to sensible values when a whole network is connected.

default: [ 0.1]

range: low = 0.0 ; high = 1.0

Iext (\(I_{ext}\))

Subcortical input strength. It represents a non-specific excitation or thalamic inputs.

default: [ 0.3]

range: low = 0.165 ; high = 0.3

QV_max (\(Q_{max}\))

Maximal firing rate for excitatory populations (kHz)

default: [ 1.]

range: low = 0.1 ; high = 1.0

QZ_max (\(Q_{max}\))

Maximal firing rate for excitatory populations (kHz)

default: [ 1.]

range: low = 0.1 ; high = 1.0

TCa (\(T_{Ca}\))

Threshold value for Ca channels.

default: [-0.01]

range: low = -0.02 ; high = -0.01

TK (\(T_{K}\))

Threshold value for K channels.

default: [ 0.]

range: low = 0.0 ; high = 0.0001

TNa (\(T_{Na}\))

Threshold value for Na channels.

default: [ 0.3]

range: low = 0.25 ; high = 0.3

VCa (\(V_{Ca}\))

Ca Nernst potential.

default: [ 1.]

range: low = 0.9 ; high = 1.1

VK (\(V_{K}\))

K Nernst potential.

default: [-0.7]

range: low = -0.8 ; high = 1.0

VL (\(V_{L}\))

Nernst potential leak channels.

default: [-0.5]

range: low = -0.7 ; high = -0.4

VNa (\(V_{Na}\))

Na Nernst potential.

default: [ 0.53]

range: low = 0.51 ; high = 0.55

VT (\(V_{T}\))

Threshold potential (mean) for excitatory neurons. In [Breaksetal_2003_b] this value is 0.

default: [ 0.]

range: low = 0.0 ; high = 0.7

ZT (\(Z_{T}\))

Threshold potential (mean) for inihibtory neurons.

default: [ 0.]

range: low = 0.0 ; high = 0.1

aee (\(a_{ee}\))

Excitatory-to-excitatory synaptic strength.

default: [ 0.4]

range: low = 0.0 ; high = 0.6

aei (\(a_{ei}\))

Excitatory-to-inhibitory synaptic strength.

default: [ 2.]

range: low = 0.1 ; high = 2.0

aie (\(a_{ie}\))

Inhibitory-to-excitatory synaptic strength.

default: [ 2.]

range: low = 0.5 ; high = 2.0

ane (\(a_{ne}\))

Non-specific-to-excitatory synaptic strength.

default: [ 1.]

range: low = 0.4 ; high = 1.0

ani (\(a_{ni}\))

Non-specific-to-inhibitory synaptic strength.

default: [ 0.4]

range: low = 0.3 ; high = 0.5

b (\(b\))

Time constant scaling factor. The original value is 0.1

default: [ 0.1]

range: low = 0.0001 ; high = 1.0

d_Ca (\(\delta_{Ca}\))

Variance of Ca channel threshold.

default: [ 0.15]

range: low = 0.1 ; high = 0.2

d_K (\(\delta_{K}\))

Variance of K channel threshold.

default: [ 0.3]

range: low = 0.1 ; high = 0.4

d_Na (\(\delta_{Na}\))

Variance of Na channel threshold.

default: [ 0.15]

range: low = 0.1 ; high = 0.2

d_V (\(\delta_{V}\))

Variance of the excitatory threshold. It is one of the main parameters explored in [Breaksetal_2003_b].

default: [ 0.65]

range: low = 0.49 ; high = 0.7

d_Z (\(\delta_{Z}\))

Variance of the inhibitory threshold.

default: [ 0.7]

range: low = 0.001 ; high = 0.75

gCa (\(g_{Ca}\))

Conductance of population of Ca++ channels.

default: [ 1.1]

range: low = 0.9 ; high = 1.5

gK (\(g_{K}\))

Conductance of population of K channels.

default: [ 2.]

range: low = 1.95 ; high = 2.05

gL (\(g_{L}\))

Conductance of population of leak channels.

default: [ 0.5]

range: low = 0.45 ; high = 0.55

gNa (\(g_{Na}\))

Conductance of population of Na channels.

default: [ 6.7]

range: low = 0.0 ; high = 10.0

phi (\(\phi\))

Temperature scaling factor.

default: [ 0.7]

range: low = 0.3 ; high = 0.9

rNMDA (\(r_{NMDA}\))

Ratio of NMDA to AMPA receptors.

default: [ 0.25]

range: low = 0.2 ; high = 0.3

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘Z’: array([-1.5, 1.5]), ‘W’: array([-1.5, 1.5]), ‘V’: array([-1.5, 1.5])}

t_scale (\(t_{scale}\))

Time scale factor

default: [ 1.]

range: low = 0.1 ; high = 1.0

tau_K (\(\tau_{K}\))

Time constant for K relaxation time (ms)

default: [ 1.]

range: low = 1.0 ; high = 10.0

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired.

default: [‘V’]

linear

Generic linear model.

tvb.simulator.models.linear.Linear

traited class Linear traits on this class:

gamma (\(\gamma\))

The damping coefficient specifies how quickly the node’s activity relaxes, must be larger than the node’s in-degree in order to remain stable.

default: [-10.]

range: low = -100.0 ; high = 0.0

state_variable_range (State Variable ranges [lo, hi])

Range used for state variable initialization and visualization.

default: {‘x’: array([-1, 1])}

variables_of_interest (Variables watched by Monitors)

default: [‘x’]

oscillator

Oscillator models.

tvb.simulator.models.oscillator.Generic2dOscillator

The Generic2dOscillator model is a generic dynamic system with two state variables. The dynamic equations of this model are composed of two ordinary differential equations comprising two nullclines. The first nullcline is a cubic function as it is found in most neuron and population models; the second nullcline is arbitrarily configurable as a polynomial function up to second order. The manipulation of the latter nullcline’s parameters allows to generate a wide range of different behaviours.

Equations:

\[\begin{split}\dot{V} &= d \, \tau (-f V^3 + e V^2 + g V + \alpha W + \gamma I), \\ \dot{W} &= \dfrac{d}{\tau}\,\,(c V^2 + b V - \beta W + a),\end{split}\]

See:

[FH_1961]

FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal 1: 445, 1961.

[Nagumo_1962]

Nagumo et.al, An Active Pulse Transmission Line Simulating Nerve Axon, Proceedings of the IRE 50: 2061, 1962.

[SJ_2011]

Stefanescu, R., Jirsa, V.K. Reduced representations of heterogeneous mixed neural networks with synaptic coupling. Physical Review E, 83, 2011.

[SJ_2010]

Jirsa VK, Stefanescu R. Neural population modes capture biologically realistic large-scale network dynamics. Bulletin of Mathematical Biology, 2010.

[SJ_2008_a]

Stefanescu, R., Jirsa, V.K. A low dimensional description of globally coupled heterogeneous neural networks of excitatory and inhibitory neurons. PLoS Computational Biology, 4(11), 2008).

The model’s (\(V\), \(W\)) time series and phase-plane its nullclines can be seen in the figure below.

The model with its default parameters exhibits FitzHugh-Nagumo like dynamics.

Table 1
EXCITABLE CONFIGURATION
Parameter	Value
a	-2.0
b	-10.0
c	0.0
d	0.02
I	0.0
limit cycle if a is 2.0

Table 2
BISTABLE CONFIGURATION
Parameter	Value
a	1.0
b	0.0
c	-5.0
d	0.02
I	0.0
monostable regime: fixed point if Iext=-2.0 limit cycle if Iext=-1.0

Table 3
EXCITABLE CONFIGURATION
(similar to Morris-Lecar)
Parameter	Value
a	0.5
b	0.6
c	-4.0
d	0.02
I	0.0
excitable regime if b=0.6 oscillatory if b=0.4

Table 4
GhoshetAl, 2008 KnocketAl, 2009
Parameter	Value
a	1.05
b	-1.00
c	0.0
d	0.1
I	0.0
alpha	1.0
beta	0.2
gamma	-1.0
e	0.0
g	1.0
f	1/3
tau	1.25
frequency peak at 10Hz

Table 5
SanzLeonetAl 2013
Parameter	Value
a	0.5
b	-10.0
c	0.0
d	0.02
I	0.0
intrinsic frequency is approx 10 Hz

NOTE: This regime, if I = 2.1, is called subthreshold regime. Unstable oscillations appear through a subcritical Hopf bifurcation.

The (\(V\), \(W\)) phase-plane for the generic 2D population model for default parameters. The dynamical system has an equilibrium point.

Generic2dOscillator.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

Generic2dOscillator.dfun(vw, c, local_coupling=0.0)

traits on this class:

I (\(I_{ext}\))

Baseline shift of the cubic nullcline

default: [ 0.]

range: low = -5.0 ; high = 5.0

a (\(a\))

Vertical shift of the configurable nullcline

default: [-2.]

range: low = -5.0 ; high = 5.0

alpha (\(\alpha\))

Constant parameter to scale the rate of feedback from the slow variable to the fast variable.

default: [ 1.]

range: low = -5.0 ; high = 5.0

b (\(b\))

Linear slope of the configurable nullcline

default: [-10.]

range: low = -20.0 ; high = 15.0

beta (\(\beta\))

Constant parameter to scale the rate of feedback from the slow variable to itself

default: [ 1.]

range: low = -5.0 ; high = 5.0

c (\(c\))

Parabolic term of the configurable nullcline

default: [ 0.]

range: low = -10.0 ; high = 10.0

d (\(d\))

Temporal scale factor. Warning: do not use it unless you know what you are doing and know about time tides.

default: [ 0.02]

range: low = 0.0001 ; high = 1.0

e (\(e\))

Coefficient of the quadratic term of the cubic nullcline.

default: [ 3.]

range: low = -5.0 ; high = 5.0

f (\(f\))

Coefficient of the cubic term of the cubic nullcline.

default: [ 1.]

range: low = -5.0 ; high = 5.0

g (\(g\))

Coefficient of the linear term of the cubic nullcline.

default: [ 0.]

range: low = -5.0 ; high = 5.0

gamma (\(\gamma\))

Constant parameter to reproduce FHN dynamics where excitatory input currents are negative. It scales both I and the long range coupling term.

default: [ 1.]

range: low = -1.0 ; high = 1.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random initial conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘W’: array([-6., 6.]), ‘V’: array([-2., 4.])}

tau (\(\tau\))

A time-scale hierarchy can be introduced for the state variables \(V\) and \(W\). Default parameter is 1, which means no time-scale hierarchy.

default: [ 1.]

range: low = 1.0 ; high = 5.0

variables_of_interest (Variables or quantities available to Monitors)

The quantities of interest for monitoring for the generic 2D oscillator.

default: [‘V’]

tvb.simulator.models.oscillator.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.

See:

[YK_1975]

Y. Kuramoto, in: H. Arakai (Ed.), International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, page 420, vol. 39, 1975.

[SS_2000]

S. H. Strogatz. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143, 2000.

[JC_2011]

J. Cabral, E. Hugues, O. Sporns, G. Deco. Role of local network oscillations in resting-state functional connectivity. NeuroImage, 57, 1, 2011.

The \(\theta\) variable is the phase angle of the oscillation.

Dynamic equations:

\[\dot{\theta}_{k} = \omega_{k} + \mathbf{\Gamma}(\theta_k, \theta_j, u_{kj}) + \sin(W_{\zeta}\theta)\]

traits on this class:

omega (\(\omega\))

\(\omega\) sets the base line frequency for the Kuramoto oscillator in [rad/ms]

default: [ 1.]

range: low = 0.01 ; high = 200.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random initial conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘theta’: array([ 0. , 6.28318531])}

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The Kuramoto model, however, only has one state variable with and index of 0, so it is not necessary to change the default here.

default: [‘theta’]

stefanescu_jirsa

Models developed by Stefanescu-Jirsa, based on reduced-set analyses of infinite populations.

tvb.simulator.models.stefanescu_jirsa.ReducedSetBase

traited class ReducedSetBase traits on this class:

tvb.simulator.models.stefanescu_jirsa.ReducedSetFitzHughNagumo

A reduced representation of a set of Fitz-Hugh Nagumo oscillators, [SJ_2008].

The models (\(\xi\), \(\eta\)) phase-plane, including a representation of the vector field as well as its nullclines, using default parameters, can be seen below:

The (\(\xi\), \(\eta\)) phase-plane for the first mode of a reduced set of Fitz-Hugh Nagumo oscillators.

The (\(\xi\), \(\eta\)) phase-plane for the second mode of a reduced set of Fitz-Hugh Nagumo oscillators.

The (\(\xi\), \(\eta\)) phase-plane for the third mode of a reduced set of Fitz-Hugh Nagumo oscillators.

ReducedSetFitzHughNagumo.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

The system’s equations for the i-th mode at node q are:

\[\begin{split}\dot{\xi}_{i} &= c\left(\xi_i-e_i\frac{\xi_{i}^3}{3} -\eta_{i}\right) + K_{11}\left[\sum_{k=1}^{o} A_{ik}\xi_k-\xi_i\right] - K_{12}\left[\sum_{k =1}^{o} B_{i k}\alpha_k-\xi_i\right] + cIE_i \\ &\, + \left[\sum_{k=1}^{o} \mathbf{\Gamma}(\xi_{kq}, \xi_{kr}, u_{qr})\right] + \left[\sum_{k=1}^{o} W_{\zeta}\cdot\xi_{kr} \right], \\ \dot{\eta}_i &= \frac{1}{c}\left(\xi_i-b\eta_i+m_i\right), \\ & \\ \dot{\alpha}_i &= c\left(\alpha_i-f_i\frac{\alpha_i^3}{3}-\beta_i\right) + K_{21}\left[\sum_{k=1}^{o} C_{ik}\xi_i-\alpha_i\right] + cII_i \\ & \, + \left[\sum_{k=1}^{o} \mathbf{\Gamma}(\xi_{kq}, \xi_{kr}, u_{qr})\right] + \left[\sum_{k=1}^{o} W_{\zeta}\cdot\xi_{kr}\right], \\ & \\ \dot{\beta}_i &= \frac{1}{c}\left(\alpha_i-b\beta_i+n_i\right),\end{split}\]

ReducedSetFitzHughNagumo.update_derived_parameters()

Calculate coefficients for the Reduced FitzHugh-Nagumo oscillator based neural field model. Specifically, this method implements equations for calculating coefficients found in the supplemental material of [SJ_2008].

Include equations here...

#NOTE: In the Article this modelis called StefanescuJirsa2D

traits on this class:

K11 (\(K_{11}\))

Internal coupling, excitatory to excitatory

default: [ 0.5]

range: low = 0.0 ; high = 1.0

K12 (\(K_{12}\))

Internal coupling, inhibitory to excitatory

default: [ 0.15]

range: low = 0.0 ; high = 1.0

K21 (\(K_{21}\))

Internal coupling, excitatory to inhibitory

default: [ 0.15]

range: low = 0.0 ; high = 1.0

a (\(a\))

doc...

default: [ 0.45]

range: low = 0.0 ; high = 1.0

b (\(b\))

doc...

default: [ 0.9]

range: low = 0.0 ; high = 1.0

mu (\(\mu\))

Mean of Gaussian distribution

default: [ 0.]

range: low = 0.0 ; high = 1.0

sigma (\(\sigma\))

Standard deviation of Gaussian distribution

default: [ 0.35]

range: low = 0.0 ; high = 1.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘alpha’: array([-4., 4.]), ‘beta’: array([-3., 3.]), ‘xi’: array([-4., 4.]), ‘eta’: array([-3., 3.])}

tau (\(\tau\))

doc...(prob something about timescale seperation)

default: [ 3.]

range: low = 1.5 ; high = 4.5

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(\xi = 0\), \(\eta = 1\), \(\alpha = 2\), and \(\beta= 3\).

default: [‘xi’, ‘alpha’]

tvb.simulator.models.stefanescu_jirsa.ReducedSetHindmarshRose

[SJ_2008]

(1, 2, 3, 4) Stefanescu and Jirsa, PLoS Computational Biology, A Low Dimensional Description of Globally Coupled Heterogeneous Neural Networks of Excitatory and Inhibitory 4, 11, 26–36, 2008.

The models (\(\xi\), \(\eta\)) phase-plane, including a representation of the vector field as well as its nullclines, using default parameters, can be seen below:

The (\(\xi\), \(\eta\)) phase-plane for the first mode of a reduced set of Hindmarsh-Rose oscillators.

The (\(\xi\), \(\eta\)) phase-plane for the second mode of a reduced set of Hindmarsh-Rose oscillators.

The (\(\xi\), \(\eta\)) phase-plane for the third mode of a reduced set of Hindmarsh-Rose oscillators.

ReducedSetHindmarshRose.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

The dynamic equations were orginally taken from [SJ_2008].

The equations of the population model for i-th mode at node q are:

\[\begin{split}\dot{\xi}_i &= \eta_i-a_i\xi_i^3 + b_i\xi_i^2- \tau_i + K_{11} \left[\sum_{k=1}^{o} A_{ik} \xi_k - \xi_i \right] - K_{12} \left[\sum_{k=1}^{o} B_{ik} \alpha_k - \xi_i\right] + IE_i \\ &\, + \left[\sum_{k=1}^{o} \mathbf{\Gamma}(\xi_{kq}, \xi_{kr}, u_{qr})\right] + \left[\sum_{k=1}^{o} W_{\zeta}\cdot\xi_{kr} \right], \\ & \\ \dot{\eta}_i &= c_i-d_i\xi_i^2 -\tau_i, \\ & \\ \dot{\tau}_i &= rs\xi_i - r\tau_i -m_i, \\ & \\ \dot{\alpha}_i &= \beta_i - e_i \alpha_i^3 + f_i \alpha_i^2 - \gamma_i + K_{21} \left[\sum_{k=1}^{o} C_{ik} \xi_k - \alpha_i \right] + II_i \\ &\, +\left[\sum_{k=1}^{o}\mathbf{\Gamma}(\xi_{kq}, \xi_{kr}, u_{qr})\right] + \left[\sum_{k=1}^{o}W_{\zeta}\cdot\xi_{kr}\right], \\ & \\ \dot{\beta}_i &= h_i - p_i \alpha_i^2 - \beta_i, \\ \dot{\gamma}_i &= rs \alpha_i - r \gamma_i - n_i,\end{split}\]

ReducedSetHindmarshRose.update_derived_parameters(corrected_d_p=True)

Calculate coefficients for the neural field model based on a Reduced set of Hindmarsh-Rose oscillators. Specifically, this method implements equations for calculating coefficients found in the supplemental material of [SJ_2008].

Include equations here...

#NOTE: In the Article this modelis called StefanescuJirsa3D

traits on this class:

K11 (\(K_{11}\))

Internal coupling, excitatory to excitatory

default: [ 0.5]

range: low = 0.0 ; high = 1.0

K12 (\(K_{12}\))

Internal coupling, inhibitory to excitatory

default: [ 0.1]

range: low = 0.0 ; high = 1.0

K21 (\(K_{21}\))

Internal coupling, excitatory to inhibitory

default: [ 0.15]

range: low = 0.0 ; high = 1.0

a (\(a\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 1.]

range: low = 0.0 ; high = 1.0

b (\(b\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 3.]

range: low = 0.0 ; high = 3.0

c (\(c\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 1.]

range: low = 0.0 ; high = 1.0

d (\(d\))

Dimensionless parameter as in the Hindmarsh-Rose model

default: [ 5.]

range: low = 2.5 ; high = 7.5

mu (\(\mu\))

Mean of Gaussian distribution

default: [ 3.3]

range: low = 1.1 ; high = 3.3

r (\(r\))

Adaptation parameter

default: [ 0.006]

range: low = 0.0 ; high = 0.1

s (\(s\))

Adaptation paramters, governs feedback

default: [ 4.]

range: low = 2.0 ; high = 6.0

sigma (\(\sigma\))

Standard deviation of Gaussian distribution

default: [ 0.3]

range: low = 0.0 ; high = 1.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘tau’: array([ 2., 10.]), ‘xi’: array([-4., 4.]), ‘beta’: array([-20., 20.]), ‘eta’: array([-25., 20.]), ‘alpha’: array([-4., 4.]), ‘gamma’: array([ 2., 10.])}

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(\xi = 0\), \(\eta = 1\), \(\tau = 2\), \(\alpha = 3\), \(\beta = 4\), and \(\gamma = 5\)

default: [‘xi’, ‘eta’, ‘tau’]

xo (\(x_{o}\))

Leftmost equilibrium point of x

default: [-1.6]

range: low = -2.4 ; high = -0.8

wilson_cowan

Wilson-Cowan equations based model definition.

tvb.simulator.models.wilson_cowan.WilsonCowan

References:

[WC_1972]

(1, 2) Wilson, H.R. and Cowan, J.D. Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical journal, 12: 1-24, 1972.

[WC_1973]

Wilson, H.R. and Cowan, J.D A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue

[D_2011]

Daffertshofer, A. and van Wijk, B. On the influence of amplitude on the connectivity between phases Frontiers in Neuroinformatics, July, 2011

Used Eqns 11 and 12 from [WC_1972] in dfun. P and Q represent external inputs, which when exploring the phase portrait of the local model are set to constant values. However in the case of a full network, P and Q are the entry point to our long range and local couplings, that is, the activity from all other nodes is the external input to the local population.

The default parameters are taken from figure 4 of [WC_1972], pag. 10

In [WC_1973] they present a model of neural tissue on the pial surface is. See Fig. 1 in page 58. The following local couplings (lateral interactions) occur given a region i and a region j:

E_i-> E_j E_i-> I_j I_i-> I_j I_i-> E_j

Table 1
SanzLeonetAl, 2014
Parameter	Value
k_e, k_i	1.00
r_e, r_i	0.00
tau_e, tau_i	10.0
c_1	10.0
c_2	6.0
c_3	1.0
c_4	1.0
a_e, a_i	1.0
b_e, b_i	0.0
theta_e	2.0
theta_i	3.5
alpha_e	1.2
alpha_i	2.0
P	0.5
Q	0
c_e, c_i	1.0
alpha_e	1.2
alpha_i	2.0
frequency peak at 20 Hz

The parameters in Table 1 reproduce Figure A1 in [D_2011] but set the limit cycle frequency to a sensible value (eg, 20Hz).

Model bifurcation parameters:

• \(c_1\)
• \(P\)

The models (\(E\), \(I\)) phase-plane, including a representation of the vector field as well as its nullclines, using default parameters, can be seen below:

The (\(E\), \(I\)) phase-plane for the Wilson-Cowan model.

WilsonCowan.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

The general formulation for the textit{textbf{Wilson-Cowan}} model as a dynamical unit at a node $k$ in a BNM with $l$ nodes reads:

\[\begin{split}\dot{E}_k &= \dfrac{1}{\tau_e} (-E_k + (k_e - r_e E_k) \mathcal{S}_e (\alpha_e \left( c_{ee} E_k - c_{ei} I_k + P_k - \theta_e + \mathbf{\Gamma}(E_k, E_j, u_{kj}) + W_{\zeta}\cdot E_j + W_{\zeta}\cdot I_j\right) ))\\ \dot{I}_k &= \dfrac{1}{\tau_i} (-I_k + (k_i - r_i I_k) \mathcal{S}_i (\alpha_i \left( c_{ie} E_k - c_{ee} I_k + Q_k - \theta_i + \mathbf{\Gamma}(E_k, E_j, u_{kj}) + W_{\zeta}\cdot E_j + W_{\zeta}\cdot I_j\right) )),\end{split}\]

traits on this class:

P (\(P\))

External stimulus to the excitatory population. Constant intensity.Entry point for coupling.

default: [ 0.]

range: low = 0.0 ; high = 20.0

Q (\(Q\))

External stimulus to the inhibitory population. Constant intensity.Entry point for coupling.

default: [ 0.]

range: low = 0.0 ; high = 20.0

a_e (\(a_e\))

The slope parameter for the excitatory response function

default: [ 1.2]

range: low = 0.0 ; high = 1.4

a_i (\(a_i\))

The slope parameter for the inhibitory response function

default: [ 1.]

range: low = 0.0 ; high = 2.0

alpha_e (\(\alpha_e\))

External stimulus to the excitatory population. Constant intensity.Entry point for coupling.

default: [ 1.]

range: low = 0.0 ; high = 20.0

alpha_i (\(\alpha_i\))

External stimulus to the inhibitory population. Constant intensity.Entry point for coupling.

default: [ 1.]

range: low = 0.0 ; high = 20.0

b_e (\(b_e\))

Position of the maximum slope of the excitatory sigmoid function

default: [ 2.8]

range: low = 1.4 ; high = 6.0

b_i (\(b_i\))

Position of the maximum slope of a sigmoid function [in threshold units]

default: [ 4.]

range: low = 2.0 ; high = 6.0

c_e (\(c_e\))

The amplitude parameter for the excitatory response function

default: [ 1.]

range: low = 1.0 ; high = 20.0

c_ee (\(c_{ee}\))

Excitatory to excitatory coupling coefficient

default: [ 12.]

range: low = 11.0 ; high = 16.0

c_ei (\(c_{ie}\))

Excitatory to inhibitory coupling coefficient.

default: [ 13.]

range: low = 2.0 ; high = 22.0

c_i (\(c_i\))

The amplitude parameter for the inhibitory response function

default: [ 1.]

range: low = 1.0 ; high = 20.0

c_ie (\(c_{ei}\))

Inhibitory to excitatory coupling coefficient

default: [ 4.]

range: low = 2.0 ; high = 15.0

c_ii (\(c_{ii}\))

Inhibitory to inhibitory coupling coefficient.

default: [ 11.]

range: low = 2.0 ; high = 15.0

k_e (\(k_e\))

Maximum value of the excitatory response function

default: [ 1.]

range: low = 0.5 ; high = 2.0

k_i (\(k_i\))

Maximum value of the inhibitory response function

default: [ 1.]

range: low = 0.0 ; high = 2.0

r_e (\(r_e\))

Excitatory refractory period

default: [ 1.]

range: low = 0.5 ; high = 2.0

r_i (\(r_i\))

Inhibitory refractory period

default: [ 1.]

range: low = 0.5 ; high = 2.0

state_variable_range (State Variable ranges [lo, hi])

The values for each state-variable should be set to encompass the expected dynamic range of that state-variable for the current parameters, it is used as a mechanism for bounding random inital conditions when the simulation isn’t started from an explicit history, it is also provides the default range of phase-plane plots.

default: {‘I’: array([ 0., 1.]), ‘E’: array([ 0., 1.])}

tau_e (\(\tau_e\))

Excitatory population, membrane time-constant [ms]

default: [ 10.]

range: low = 0.0 ; high = 150.0

tau_i (\(\tau_i\))

Inhibitory population, membrane time-constant [ms]

default: [ 10.]

range: low = 0.0 ; high = 150.0

theta_e (\(\theta_e\))

Excitatory threshold

default: [ 0.]

range: low = 0.0 ; high = 60.0

theta_i (\(\theta_i\))

Inhibitory threshold

default: [ 0.]

range: low = 0.0 ; high = 60.0

variables_of_interest (Variables watched by Monitors)

This represents the default state-variables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding state-variable indices for this model are \(E = 0\) and \(I = 1\).

default: [‘E’]

wong_wang

Models based on Wong-Wang’s work.

tvb.simulator.models.wong_wang.ReducedWongWang

[WW_2006]

Kong-Fatt Wong and Xiao-Jing Wang, A Recurrent Network Mechanism of Time Integration in Perceptual Decisions. Journal of Neuroscience 26(4), 1314-1328, 2006.

[DPA_2013]

Deco Gustavo, Ponce Alvarez Adrian, Dante Mantini, Gian Luca Romani, Patric Hagmann and Maurizio Corbetta. Resting-State Functional Connectivity Emerges from Structurally and Dynamically Shaped Slow Linear Fluctuations. The Journal of Neuroscience 32(27), 11239-11252, 2013.

ReducedWongWang.__init__()

x.__init__(...) initializes x; see help(type(x)) for signature

Equations taken from [DPA_2013] , page 11242

\[\begin{split}x_k &= w\,J_N \, S_k + I_o + J_N \mathbf\Gamma(S_k, S_j, u_{kj}),\\ H(x_k) &= \dfrac{ax_k - b}{1 - \exp(-d(ax_k -b))},\\ \dot{S}_k &= -\dfrac{S_k}{\tau_s} + (1 - S_k) \, H(x_k) \, \gamma\end{split}\]

traits on this class:

I_o (\(I_{o}\))

[nA] Effective external input

default: [ 0.33]

range: low = 0.0 ; high = 1.0

J_N (\(J_{N}\))

Excitatory recurrence

default: [ 0.2609]

range: low = 0.2609 ; high = 0.5

a (\(a\))

[n/C]. Input gain parameter, chosen to fit numerical solutions.

default: [ 0.27]

range: low = 0.0 ; high = 0.27

b (\(b\))

[kHz]. Input shift parameter chosen to fit numerical solutions.

default: [ 0.108]

range: low = 0.0 ; high = 1.0

d (\(d\))

[ms]. Parameter chosen to fit numerical solutions.

default: [ 154.]

range: low = 0.0 ; high = 200.0

gamma (\(\gamma\))

Kinetic parameter

default: [ 0.641]

range: low = 0.0 ; high = 1.0

sigma_noise (\(\sigma_{noise}\))

[nA] Noise amplitude. Take this value into account for stochatic integration schemes.

default: [ 1.00000000e-09]

range: low = 0.0 ; high = 0.005

state_variable_range (State variable ranges [lo, hi])

Population firing rate

default: {‘S’: array([ 0., 1.])}

tau_s (\(\tau_S\))

Kinetic parameter. NMDA decay time constant.

default: [ 100.]

range: low = 50.0 ; high = 150.0

variables_of_interest (Variables watched by Monitors)

default state variables to be monitored

default: [‘S’]

w (\(w\))

Excitatory recurrence

default: [ 0.6]

range: low = 0.0 ; high = 1.0