Python codes for Spikes, decisions, and actions, Wilson 1999

Installation, How to use

• using on Colab (Recommended)

  • Go to examples

  • Open a notebook and click on “open on colab”

  • Uncomment the cell with pip install command to install the netsci package.

• using on local machines

pip3 install -e .
# or
pip install "git+https://github.com/Ziaeemehr/spikes.git"

Indices and tables

• Index

• Module Index

• Search Page

Chapters

Contents:

• Chapter 3
  • Two-dimensional systems and state space
  • Analytical Solution for \(\frac{d\vec{X}}{dt} = A\vec{X} + \vec{B}\)
    • Steps to Solve Analytically
  • Numerical solution
  • Analytical solution
  • Direction Fields
• Chapter 4
  • Higher dimensional linear systems
    • Eq. 4.2
    • 4.3 Oscillatory control of respiration
• Chapter 5
  • Approximation and simulation
    • Exercise 2, Solving with Euler method
    • Exercise 1.
• Chapter 6
  • Nonlinear neurodynamics and bifurcations
  • Step 1: Define the System of Differential Equations
  • Step 2: Find the Nullclines
  • Step 3: Use SymPy to Solve for Nullclines
  • Step 4: Plot the Nullclines Using Matplotlib
  • Example
  • Explanation
  • Tools Used
    • adding an interactive trajectory solution
    • Related tool
  • pplane
  • Stabiliby of steady states
    • Explanation:
    • Customizing the Code:
  • A short-term memory circuit
  • Hysteresis, bifurcation, and memory
  • Adaptation, forgetting, and catastrophe theory
  • Competition and neural decisions

API and documentation

spikes.utils

characteristic_polynomial(matrix)

Computes the characteristic polynomial of a given matrix.

Parameters:

matrix: (list of lists)

A 2D list representing the matrix.

Returns:

sympy.Poly:

The characteristic polynomial of the matrix.

routh(p)

Construct the Routh-Hurwitz array given a polynomial in s

Parameters:

p: sympy.Poly

The characteristic polynomial of coefficient matrix

Returns:

value: sympy.Matrix

The Routh-Hurwitz array

References https://github.com/alchemyst/Dynamics-and-Control/blob/master/tbcontrol/symbolic.py

spikes.plot

plot_direction_field(A: ndarray, B: ndarray, x1_range: tuple = (-2, 2), x2_range: tuple = (-2, 2), num_points: int = 20, ax=None, xlabel: str = 'x1', ylabel: str = 'x2', title: str = 'Direction Field for the System $dX/dt = AX + B$', **kwargs)

Plots the direction field for the system dx/dt = A * X + B.

Parameters:

Anumpy.ndarray

A 2x2 numpy array representing the coefficient matrix for the linear system.

Bnumpy.ndarray

A 1x2 numpy array representing the constant vector.

x1_rangetuple, optional

The range for x1 values. Default is (-10, 10).

x2_rangetuple, optional

The range for x2 values. Default is (-10, 10).

num_pointsint, optional

The number of points per axis in the grid. Default is 20.

Returns:

axmatplotlib.axes.Axes

The axis object with the direction field plotted.

plot_nullclines(f, g, symbols: List[str], x_range=(-2, 2), y_range=(-2, 2), num_points=400, figsize: tuple = (4, 4), title: str = 'Nullclines of the system', grid=False, ax=None, loc='best', **kwargs: dict)

Plots the nullclines of a two-dimensional system of differential equations.

Parameters:

fsympy expression

The right-hand side of the first differential equation dx/dt = f(x, y).

gsympy expression

The right-hand side of the second differential equation dy/dt = g(x, y).

x_rangetuple, optional

The range of x values for plotting (default is (-2, 2)).

y_rangetuple, optional

The range of y values for plotting (default is (-2, 2)).

num_pointsint, optional

The number of points to plot (default is 400).

loc: string, optional

The location of the legend (default is ‘best’).

**kwargsdict

Additional keyword arguments to be passed to plt.plot.

Returns:

axmatplotlib.axes.Axes

The axis object with the nullclines plotted.

spikes.solver

solve_system_of_equations(A_np: ndarray, B_np: ndarray, X0_np: ndarray, t_range: ndarray, t0: float = 0, verbose: bool = True)

Solve a system of first-order linear differential equations of any size.

Parameters:

• A_np (ndarray) – Coefficient matrix A.

• B_np (ndarray) – Constant vector B.

• X0_np (ndarray) – Initial condition vector X0.

• t_range (array-like) – Range of time points over which to evaluate the solution.

• t0 (float) – Initial time, default is 0.

Returns:

A tuple containing: - final_solution (list): List of symbolic solutions. - x_values (list): List of evaluated solution values over t_range.

Return type:

tuple

solve_linear_system_numerical(A, B, X0, t)

Solves the differential equation dX/dt = AX + B.

Parameters:

• A (numpy.ndarray) – Coefficient matrix.

• B (numpy.ndarray) – Constant vector.

• X0 (numpy.ndarray) – Initial condition vector.

• t (numpy.ndarray) – Array of time points at which to solve.

Returns:

Array of solution vectors at each time point.

Return type:

numpy.ndarray

solve_linear_system_analytically(A, B, X0, t)

Depricated

solve_linear_system_sympy(A, B, X0, verbose=True)

Depricated