The ODE toolbox is a framework for the automatic symbolic analysis of the differential equations used for modeling spiking neuron models.

Disclaimer: Although the toolbox is hosted under the GitHub account of the NEST Initiative, it is completely independent of any specific simulator. It was, however, initially developed in the context of the NESTML project, in which the main focus was on the class of neurons presently available in the NEST simulator.

The ode-toolbox requires SymPy in a version >= 1.1.1, which can be installed using pip install sympy. The stiffness tester depends on an an installation of PyGSL. If PyGSL is not installed, the test for stiffness is skipped during the analysis of the equations (the remaining analysis is still performed).

For details, see the file requirements.txt.

To install the framework, use the following commands in a terminal:

python setup.py install

If you want to install the ODE toolbox into your home directory, add the option --user to the above call.

To run the unit and integration tests that come with the ODE toolbox, you can run the following command:

python setup.py test

Please note that this requires the pytest package to be installed.

The ode-toolbox can be used in two ways:

1. as a Python module. See the tests for examples of the exact usage of the functions.
2. as command line application. In this case, the input is stored in a json file, whose file format will be explained in the next section. The command line invocation in this case looks like this:

ode_analyzer.py <json_file>

Input for ode_analyzer.py JSON files are composed of two lists of dictionaries shapes and odes and a dictionary parameters. shapes and odes must be stated at any call of the ode_analyzer.py. parameters dictionary is stated only if a stiffness test should be performed.

Odes-list contains dictionaries each of which specifies an ODE with initial values. Every dictionary has the following keys:

• symbol: The unambiguous name of the shape
• definition: An arbitrary Python expression with free variables (an arbitrary valid Python variable name, e.g. V_m), derivative-variables (an arbitrary valid Python variable name with a postfix of a sequence of '-characters, e.g. g_in''') and functions from the math-package. The definition of a function must depend on t-variable.
• upper_bound: defines an optional condition as a valid boolean Python expression for the maximal value of the symbol-variable. This bound is used in the stiffness test.
• lower_bound: defines an optional condition as a valid boolean Python expression for the minimal value of symbol-variable. This bound is used in the stiffness test.
• initial_values: A list with Python expressions which define an initial value for every order of the shape-ODE. The length of this list defines the order of the corresponding ODE. initial_values must be stated only for the ode-shape.

Model parameters and their values are given in the parameters dictionary. This dictionary maps default values to parameter names and has to contain an entry for each free variable occurring in the equations of the odes or shapes. Every dictionary entry hat the name of the free variable as its key and a valid Python expression as its value. I.e. variable_name: expression.

Shapes-list contains shapes which can be selectively specified as a function of time (by referencing a predefined variable t) or as an ODE with initial conditions. Every shape is a dictionary with the following keys:

• type: determines whether the shape is given as a function of time (function) or as an ODE with initial conditions (ode)
• symbol: The unambiguous name of the shape
• definition: An arbitrary Python expression with free variables (an arbitrary valid Python variable name, e.g. V_m), derivative-variables (an arbitrary valid Python variable name with a postfix of a sequence of '-characters, e.g. g_in''') and functions from the math-package. The definition of a function must depend on t-variable.
• initial_values: A list with Python expressions which define an initial value for every order of the shape-ODE. The length of this list defines the order of the corresponding ODE. initial_values must be stated only for the ode-shape.

The analysis output is returned in the form of a Python dictionary. If the input was given as JSON in a file input.json, the output is stored in a file input_reault.json in the current working directory.

• solver: states which integration scheme was selected. Possible values are analytical or numerical .
  • if the stiffness tester was run (i.e. if PyGSL is available), this field also contains either explicit of implicit after a dash (-).
• ode_updates: contains a list of Python statements for updating the ODE.
• shape_state_variables: is a list of the names of the shapes and their derivatives up to the order of the ODE they satisfy minus one.
• shape_initial_values: are the initial values of the differential equations the shapes satisfy.
• shape_state_updates: contains a list of instructions to be applied to shape_state_variables in every update step if the ODE is solved analytically.
• propagator: contains a list of all non-zero entries of all propagator matrices numbered according to the order in which they should be executed if the ODE is solved analytically.
• ode_system_type: states if the current ode system with provided parameters is a stiff and non-stiff problem. Possible values stiff, non-stiff, skipped (if due to missing PyGSL or parameters the test was not performed)

The following example shows an input model that corresponds to a current-based integrate-and-fire neuron with an alpha-shaped postsynaptic response. The model ODEs can be solved analytically.

{
  "shapes": [
    {
      "type": "function",
      "symbol": "I_shape_in",
      "definition": "(e/tau_syn_in) * t * exp(-t/tau_syn_in)"
    },
    {
      "type": "ode",
      "symbol": "I_shape_ex",
      "definition": "(-1)/(tau_syn_ex)**(2)*I_shape_ex+(-2)/tau_syn_ex*I_shape_ex'",
      "initial_values": ["0",  "e / tau_syn_ex"]
    }
  ],

  "odes": [
    {
      "symbol": "V_abs",
      "definition": "(-1)/Tau*V_abs+1/C_m*(I_shape_in+I_shape_ex+I_e+currents)"
    }
  ]

}

Output:

{
  "ode_updates": [
    "V_abs = (exp(-__h/Tau)) * V_abs+ ((I_e + currents)/C_m) * (Tau - Tau*exp(-__h/Tau))", 
    "V_abs += I_shape_in*__P_I_shape_in__2_1 + I_shape_in__d*__P_I_shape_in__2_0", 
    "V_abs += I_shape_ex*__P_I_shape_ex__2_1 + I_shape_ex__d*__P_I_shape_ex__2_0"
  ], 
  "solver": "analytical", 
  "shape_initial_values": [
    [
      "0", 
      "e/tau_syn_in"
    ], 
    [
      "0", 
      "e/tau_syn_ex"
    ]
  ], 
  "shape_state_updates": [
    [
      "I_shape_in__d*__P_I_shape_in__0_0", 
      "I_shape_in*__P_I_shape_in__1_1 + I_shape_in__d*__P_I_shape_in__1_0"
    ], 
    [
      "I_shape_ex__d*__P_I_shape_ex__0_0", 
      "I_shape_ex*__P_I_shape_ex__1_1 + I_shape_ex__d*__P_I_shape_ex__1_0"
    ]
  ], 
  "shape_state_variables": [
    [
      "I_shape_in__d", 
      "I_shape_in"
    ], 
    [
      "I_shape_ex__d", 
      "I_shape_ex"
    ]
  ], 
  "propagator": {
    "__P_I_shape_in__1_1": "exp(-__h/tau_syn_in)", 
    "__P_I_shape_in__1_0": "__h*exp(-__h/tau_syn_in)", 
    "__P_I_shape_in__0_0": "exp(-__h/tau_syn_in)", 
    "__P_I_shape_ex__2_2": "exp(-__h/Tau)", 
    "__P_I_shape_ex__2_0": "Tau*tau_syn_ex*(Tau*tau_syn_ex*exp(2*__h/tau_syn_ex) - Tau*tau_syn_ex*exp(__h*(Tau + tau_syn_ex)/(Tau*tau_syn_ex)) - __h*(Tau - tau_syn_ex)*exp(__h*(Tau + tau_syn_ex)/(Tau*tau_syn_ex)))*exp(-__h*(2*Tau + tau_syn_ex)/(Tau*tau_syn_ex))/(C_m*(Tau - tau_syn_ex)**2)", 
    "__P_I_shape_ex__2_1": "Tau*tau_syn_ex*(-exp(__h/Tau) + exp(__h/tau_syn_ex))*exp(-__h*(Tau + tau_syn_ex)/(Tau*tau_syn_ex))/(C_m*(Tau - tau_syn_ex))", 
    "__P_I_shape_in__2_0": "Tau*tau_syn_in*(Tau*tau_syn_in*exp(2*__h/tau_syn_in) - Tau*tau_syn_in*exp(__h*(Tau + tau_syn_in)/(Tau*tau_syn_in)) - __h*(Tau - tau_syn_in)*exp(__h*(Tau + tau_syn_in)/(Tau*tau_syn_in)))*exp(-__h*(2*Tau + tau_syn_in)/(Tau*tau_syn_in))/(C_m*(Tau - tau_syn_in)**2)", 
    "__P_I_shape_in__2_1": "Tau*tau_syn_in*(-exp(__h/Tau) + exp(__h/tau_syn_in))*exp(-__h*(Tau + tau_syn_in)/(Tau*tau_syn_in))/(C_m*(Tau - tau_syn_in))", 
    "__P_I_shape_in__2_2": "exp(-__h/Tau)", 
    "__P_I_shape_ex__1_1": "exp(-__h/tau_syn_ex)", 
    "__P_I_shape_ex__0_0": "exp(-__h/tau_syn_ex)", 
    "__P_I_shape_ex__1_0": "__h*exp(-__h/tau_syn_ex)"
  }
}

The following example shows an input model that corresponds to a conductance-based integrate-and-fire neuron with an alpha-shaped postsynaptic response. This example provides also a parameters dictionary for the stiffness testing.

{
  "shapes": [
    {
      "type": "function",
      "symbol": "g_in",
      "definition": "(e/tau_syn_in)*t*exp((-1)/tau_syn_in*t)"
    },
    {
      "type": "ode",
      "symbol": "g_ex",
      "definition": "(-1)/(tau_syn_ex)**(2)*g_ex+(-2)/tau_syn_ex*g_ex'",
      "initial_values": ["0",  "e / tau_syn_ex"]
    }
  ],

  "odes": [
    {
      "symbol": "V_m",
      "definition": "(-(g_L*(V_m-E_L))-(g_ex*(V_m-E_ex))-(g_in*(V_m-E_in))+I_stim+I_e)/C_m",
      "initial_values": ["E_L"],
      "upper_bound": "V_th"
    }
  ],

  "parameters": {
    "V_th": "-55.0",
    "g_L": "16.6667",
    "C_m": "250.0",
    "E_ex": "0",
    "E_in": "-85.0",
    "E_L": "-70.0",
    "tau_syn_ex": "0.02",
    "tau_syn_in": "2.0",
    "I_e": "0",
    "I_stim": "0"
  }
}

Output:

{
  "shape_state_variables": [
    "g_in__d", 
    "g_in", 
    "g_ex__d", 
    "g_ex"
  ], 
  "shape_initial_values": [
    "0", 
    "e/tau_syn_in", 
    "0", 
    "e/tau_syn_ex"
  ], 
  "shape_ode_definitions": [
    "-1/tau_syn_in**2 * g_in + -2/tau_syn_in * g_in__d", 
    "-1/tau_syn_ex**2 * g_ex + -2/tau_syn_ex * g_ex__d"
  ], 
  "solver": "numeric",
  "ode_system_type": "stiff"
}