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.
This module defines the common imports and abstract base class for model definitions.
HindmarshRoseJirsa Epileptor model.
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., 1e3, 1e3, 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. 
x.__init__(...) initializes x; see help(type(x)) for signature
Variables of interest to be used by monitors: y[0] + y[3]
\[\begin{split}\dot{x_{1}} &=& y_{1}  f_{1}(x_{1}, x_{2})  z + I_{ext1} \\ \dot{y_{1}} &=& c  d x_{1}^{2}  y{1} \\ \dot{z} &=& \begin{cases} r(4 (x_{1}  x_{0})  z0.1 z^{7}) & \text{if } x<0 \\ r(4 (x_{1}  x_{0})  z) & \text{if } x \geq 0 \end{cases} \\ \dot{x_{2}} &=& y_{2} + x_{2}  x_{2}^{3} + I_{ext2} + 0.002 g  0.3 (z3.5) \\ \dot{y_{2}} &=& 1 / \tau (y_{2} + f_{2}(x_{2}))\\ \dot{g} &=& 0.01 (g  0.1 x_{1})\end{split}\]
and:
\[\begin{split}f_{2}(x_{2}) = \begin{cases} 0 & \text{if } x_{2} <0.25\\ a_{2}(x_{2} + 0.25) & \text{if } x_{2} \geq 0.25 \end{cases}\end{split}\]
traits on this class:
 Iext (Iext)
External input current to the first populationdefault: [ 3.1]range: low = 1.5 ; high = 5.0 Iext2 (Iext2)
External input current to the second populationdefault: [ 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 scaledefault: [ 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 variabledefault: [1] aa (aa)
Linear coefficient in fifth state variabledefault: [6] b (b)
Coefficient of the squared term in the first state variabeldefault: [3] c (c)
Additive coefficient for the second state variable, called \(y_{0}\) in Jirsa paperdefault: [1] d (d)
Coefficient of the squared term in the second state variabledefault: [5] r (r)
Temporal scaling in the third state variable, called \(1/\tau_{0}\) in Jirsa paperdefault: [ 0.00035]range: low = 0.0 ; high = 0.001 s (s)
Linear coefficient in the third state variabledefault: [4] slope (slope)
Linear coefficient in the first state variabledefault: [ 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 variabledefault: [10] tt (tt)
Time scaling of the whole systemdefault: [ 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 parameterdefault: [1.6]range: low = 3.0 ; high = 1.0
Hopfield model with modifications following Golos & Daucé.
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 spinglass 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, 25542558, 1982.
[Hopfield1984] Hopfield, J. J., Neurons with graded response have collective computational properties like those of twosate neurons, Proc. Nat. Acad. Sci. (USA) 81, 30883092, 1984.
See also, http://www.scholarpedia.org/article/Hopfield_network
x.__init__(...) initializes x; see help(type(x)) for signature
Dynamic equations:
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 statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane plots.default: {‘x’: array([1., 2.]), ‘theta’: array([ 0., 1.])} tauT (\(\tau_{\theta}\))
The slow timescale for threshold calculus :math:` heta`, statevariable of the model.default: [ 5.]range: low = 0.01 ; high = 100.0 taux (\(\tau_{x}\))
The fast timescale for potential calculus \(x\), statevariable of the model.default: [ 1.]range: low = 0.01 ; high = 100.0 variables_of_interest (Variables watched by Monitors)
The values for each statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane plots.default: [‘x’]
JansenRit and derivative models.
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 neurophysiologicallybased mathematical model of flash visual evoked potentials 
x.__init__(...) initializes x; see help(type(x)) for signature
The dynamic equations were taken from [JR_1995]
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 ratedefault: [ 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 statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane 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 statevariables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding statevariable indices for this model are \(y0 = 0\), \(y1 = 1\), \(y2 = 2\), \(y3 = 3\), \(y4 = 4\), and \(y5 = 5\)default: [‘y0’, ‘y1’, ‘y2’, ‘y3’]
Zetterberg et al derived a model inspired by the WilsonCowan equations. It served as a basis for the later, better known JansenRit model.
[ZL_1978]  Zetterberg LH, Kristiansson L and Mossberg K. Performance of a Model for a Local Neuron Population. Biological Cybernetics 31, 1526, 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 neurophysiologicallybased 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 statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane 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 statevariables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding statevariable 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’]
LarterBreakspear model based on the MorrisLecar equations.
A modified MorrisLecar 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: 703732, 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 restingstate functional connectivity from structural connectivity. (2009), PNAS, 106, 20352040 
[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 nondimensional 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 longrange 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.
x.__init__(...) initializes x; see help(type(x)) for signature
Dynamic equations:
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 selfcoupling. 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 nonspecific 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 (\(a_{ee}\))
Excitatorytoexcitatory synaptic strength.default: [ 0.4]range: low = 0.0 ; high = 0.6 aei (\(a_{ei}\))
Excitatorytoinhibitory synaptic strength.default: [ 2.]range: low = 0.1 ; high = 2.0 aie (\(a_{ie}\))
Inhibitorytoexcitatory synaptic strength.default: [ 2.]range: low = 0.5 ; high = 2.0 ane (\(a_{ne}\))
Nonspecifictoexcitatory synaptic strength.default: [ 1.]range: low = 0.4 ; high = 1.0 ani (\(a_{ni}\))
Nonspecifictoinhibitory synaptic strength.default: [ 0.4]range: low = 0.3 ; high = 0.5 b (\(b\))
Time constant scaling factor. The original value is 0.1default: [ 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 statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane 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 factordefault: [ 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 statevariables of this Model to be monitored. It can be overridden for each Monitor if desired.default: [‘V’]
Generic linear model.
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 indegree 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 models.
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:
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 largescale 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 phaseplane its nullclines can be seen in the figure below.
The model with its default parameters exhibits FitzHughNagumo 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 MorrisLecar)  
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 

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 (\(V\), \(W\)) phaseplane for the generic 2D population model for default parameters. The dynamical system has an equilibrium point.
x.__init__(...) initializes x; see help(type(x)) for signature
traits on this class:
 I (\(I_{ext}\))
Baseline shift of the cubic nullclinedefault: [ 0.]range: low = 5.0 ; high = 5.0 a (\(a\))
Vertical shift of the configurable nullclinedefault: [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 nullclinedefault: [10.]range: low = 20.0 ; high = 15.0 beta (\(\beta\))
Constant parameter to scale the rate of feedback from the slow variable to itselfdefault: [ 1.]range: low = 5.0 ; high = 5.0 c (\(c\))
Parabolic term of the configurable nullclinedefault: [ 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 statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane plots.default: {‘W’: array([6., 6.]), ‘V’: array([2., 4.])} tau (\(\tau\))
A timescale hierarchy can be introduced for the state variables \(V\) and \(W\). Default parameter is 1, which means no timescale 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’]
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 restingstate functional connectivity. NeuroImage, 57, 1, 2011.
The \(\theta\) variable is the phase angle of the oscillation.
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 statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane plots.default: {‘theta’: array([ 0. , 6.28318531])} variables_of_interest (Variables watched by Monitors)
This represents the default statevariables 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’]
Models developed by StefanescuJirsa, based on reducedset analyses of infinite populations.
traited class ReducedSetBase traits on this class:
A reduced representation of a set of FitzHugh Nagumo oscillators, [SJ_2008].
The models (\(\xi\), \(\eta\)) phaseplane, including a representation of the vector field as well as its nullclines, using default parameters, can be seen below:
x.__init__(...) initializes x; see help(type(x)) for signature
The system’s equations for the ith mode at node q are:
Calculate coefficients for the Reduced FitzHughNagumo 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 excitatorydefault: [ 0.5]range: low = 0.0 ; high = 1.0 K12 (\(K_{12}\))
Internal coupling, inhibitory to excitatorydefault: [ 0.15]range: low = 0.0 ; high = 1.0 K21 (\(K_{21}\))
Internal coupling, excitatory to inhibitorydefault: [ 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 distributiondefault: [ 0.]range: low = 0.0 ; high = 1.0 sigma (\(\sigma\))
Standard deviation of Gaussian distributiondefault: [ 0.35]range: low = 0.0 ; high = 1.0 state_variable_range (State Variable ranges [lo, hi])
The values for each statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane 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 statevariables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding statevariable indices for this model are \(\xi = 0\), \(\eta = 1\), \(\alpha = 2\), and \(\beta= 3\).default: [‘xi’, ‘alpha’]
[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 (\(\xi\), \(\eta\)) phaseplane, including a representation of the vector field as well as its nullclines, using default parameters, can be seen below:
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 ith mode at node q are:
Calculate coefficients for the neural field model based on a Reduced set of HindmarshRose 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 excitatorydefault: [ 0.5]range: low = 0.0 ; high = 1.0 K12 (\(K_{12}\))
Internal coupling, inhibitory to excitatorydefault: [ 0.1]range: low = 0.0 ; high = 1.0 K21 (\(K_{21}\))
Internal coupling, excitatory to inhibitorydefault: [ 0.15]range: low = 0.0 ; high = 1.0 a (\(a\))
Dimensionless parameter as in the HindmarshRose modeldefault: [ 1.]range: low = 0.0 ; high = 1.0 b (\(b\))
Dimensionless parameter as in the HindmarshRose modeldefault: [ 3.]range: low = 0.0 ; high = 3.0 c (\(c\))
Dimensionless parameter as in the HindmarshRose modeldefault: [ 1.]range: low = 0.0 ; high = 1.0 d (\(d\))
Dimensionless parameter as in the HindmarshRose modeldefault: [ 5.]range: low = 2.5 ; high = 7.5 mu (\(\mu\))
Mean of Gaussian distributiondefault: [ 3.3]range: low = 1.1 ; high = 3.3 r (\(r\))
Adaptation parameterdefault: [ 0.006]range: low = 0.0 ; high = 0.1 s (\(s\))
Adaptation paramters, governs feedbackdefault: [ 4.]range: low = 2.0 ; high = 6.0 sigma (\(\sigma\))
Standard deviation of Gaussian distributiondefault: [ 0.3]range: low = 0.0 ; high = 1.0 state_variable_range (State Variable ranges [lo, hi])
The values for each statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane 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 statevariables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding statevariable 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 xdefault: [1.6]range: low = 2.4 ; high = 0.8
WilsonCowan equations based model definition.
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: 124, 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).
The models (\(E\), \(I\)) phaseplane, including a representation of the vector field as well as its nullclines, using default parameters, can be seen below:
x.__init__(...) initializes x; see help(type(x)) for signature
The general formulation for the textit{textbf{WilsonCowan}} model as a dynamical unit at a node $k$ in a BNM with $l$ nodes reads:
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 functiondefault: [ 1.2]range: low = 0.0 ; high = 1.4 a_i (\(a_i\))
The slope parameter for the inhibitory response functiondefault: [ 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 functiondefault: [ 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 functiondefault: [ 1.]range: low = 1.0 ; high = 20.0 c_ee (\(c_{ee}\))
Excitatory to excitatory coupling coefficientdefault: [ 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 functiondefault: [ 1.]range: low = 1.0 ; high = 20.0 c_ie (\(c_{ei}\))
Inhibitory to excitatory coupling coefficientdefault: [ 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 functiondefault: [ 1.]range: low = 0.5 ; high = 2.0 k_i (\(k_i\))
Maximum value of the inhibitory response functiondefault: [ 1.]range: low = 0.0 ; high = 2.0 r_e (\(r_e\))
Excitatory refractory perioddefault: [ 1.]range: low = 0.5 ; high = 2.0 r_i (\(r_i\))
Inhibitory refractory perioddefault: [ 1.]range: low = 0.5 ; high = 2.0 state_variable_range (State Variable ranges [lo, hi])
The values for each statevariable should be set to encompass the expected dynamic range of that statevariable 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 phaseplane plots.default: {‘I’: array([ 0., 1.]), ‘E’: array([ 0., 1.])} tau_e (\(\tau_e\))
Excitatory population, membrane timeconstant [ms]default: [ 10.]range: low = 0.0 ; high = 150.0 tau_i (\(\tau_i\))
Inhibitory population, membrane timeconstant [ms]default: [ 10.]range: low = 0.0 ; high = 150.0 theta_e (\(\theta_e\))
Excitatory thresholddefault: [ 0.]range: low = 0.0 ; high = 60.0 theta_i (\(\theta_i\))
Inhibitory thresholddefault: [ 0.]range: low = 0.0 ; high = 60.0 variables_of_interest (Variables watched by Monitors)
This represents the default statevariables of this Model to be monitored. It can be overridden for each Monitor if desired. The corresponding statevariable indices for this model are \(E = 0\) and \(I = 1\).default: [‘E’]
Models based on WongWang’s work.
[WW_2006]  KongFatt Wong and XiaoJing Wang, A Recurrent Network Mechanism of Time Integration in Perceptual Decisions. Journal of Neuroscience 26(4), 13141328, 2006. 
[DPA_2013]  Deco Gustavo, Ponce Alvarez Adrian, Dante Mantini, Gian Luca Romani, Patric Hagmann and Maurizio Corbetta. RestingState Functional Connectivity Emerges from Structurally and Dynamically Shaped Slow Linear Fluctuations. The Journal of Neuroscience 32(27), 1123911252, 2013. 
x.__init__(...) initializes x; see help(type(x)) for signature
Equations taken from [DPA_2013] , page 11242
traits on this class:
 I_o (\(I_{o}\))
[nA] Effective external inputdefault: [ 0.33]range: low = 0.0 ; high = 1.0 J_N (\(J_{N}\))
Excitatory recurrencedefault: [ 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 parameterdefault: [ 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: [ 1.00000000e09]range: low = 0.0 ; high = 0.005 state_variable_range (State variable ranges [lo, hi])
Population firing ratedefault: {‘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 monitoreddefault: [‘S’] w (\(w\))
Excitatory recurrencedefault: [ 0.6]range: low = 0.0 ; high = 1.0