[WIP] Resting state integration

[vigneswaran-chandrasekaran]

This PR is the newer version of PR #1123
To follow up please refer to the older one ( #1123 ) conversations for the discussions and comments.

Brief description:

Calculation of resting-state of the model, using the resting_state function of NeuronGroup class.

Usage syntax:

group = NeuronGroup(...)
equilibrium_point = group.resting_state(initial_guess)

Simple example:

tau = 10*ms
eqs = '''dv/dt = (1-v)/tau : 1'''
group = NeuronGroup(1, eqs, method='exact')
result = group.resting_state({'v':  0})    # giving initial guess

Fixes #1064

[vigneswaran-chandrasekaran]

I've made the necessary changes for integrating Jacobian to calculate resting state as discussed previously. The new structure now looks like,

• The NeuronGroup's member function resting_state() prepares the initial guess of state variables (default to zero if not given by the user).
• Further, resting_state() calls two helper functions, _rhs_equation and _jacobian_equation, which returns the NeuronGroup objects with RHS and Jacobian equations respectively.
• rhs_group and jac_group acts as bare-skeleton NeuronGroup objects with defined governing equations. The above groups are passed to _wrapper().
• _wrapper() calls evaluate_states(group, values, states) function, which sets/gets the values/states of the corresponding group (i.e. rhs_group or jac_group)
• Once, the solution is found by root(), the Jacobin matrix is prepared for the solution, using the same helper functions and stability is analyzed

Everything works fine and solutions are accurate, I also added a test case for stability check. However, a couple of doubts including,

• The fjac of the result of root() is different from actual Jacobin of the result
• Number of iter with jac = True and jac = False are not much different

Edit 1:

• I found the definition of fjac in a page as,

the orthogonal matrix, q, produced by the QR factorization of the final approximate Jacobian matrix, stored column wise