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Signed-off-by: David Rotermund <54365609+davrot@users.noreply.github.com>
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@ -170,70 +170,6 @@ result = sympy.dsolve(diffeq, f(x))
print(result) # -> Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2) print(result) # -> Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)
``` ```
Sometimes it is necessary to simplyify the problem for the solver:
```python
import sympy
u = sympy.symbols("u", cls=sympy.Function, real=True)
t, tau, a0, urest, uc, r, i, uthr = sympy.symbols(
"t tau a0 urest uc r i uthr", real=True
)
c0, c1, c2 = sympy.symbols("c0 c1 c2", real=True)
diffeq_rhs = (a0 * (u(t) - urest) * (u(t) - uc) + r * i) / tau
print("Original")
print(diffeq_rhs)
print()
diffeq_rhs = sympy.expand(diffeq_rhs)
diffeq_rhs = sympy.collect(diffeq_rhs, u(t))
c0_coeffs = diffeq_rhs.coeff(u(t), 0)
c1_coeffs = diffeq_rhs.coeff(u(t), 1)
c2_coeffs = diffeq_rhs.coeff(u(t), 2)
diffeq_rhs = diffeq_rhs.subs(c0_coeffs, c0)
diffeq_rhs = diffeq_rhs.subs(c1_coeffs, c1)
diffeq_rhs = diffeq_rhs.subs(c2_coeffs, c2)
print("Simplified")
print(diffeq_rhs)
print()
diffeq = sympy.Eq(u(t).diff(t), diffeq_rhs)
print("Full equation")
print(diffeq)
print()
solved_diffeq_rhs = sympy.dsolve(diffeq, u(t), ics={u(0): uc}, simplify=False).rhs
solved_diffeq_rhs = solved_diffeq_rhs.subs(c0, c0_coeffs)
solved_diffeq_rhs = solved_diffeq_rhs.subs(c1, c1_coeffs)
solved_diffeq_rhs = solved_diffeq_rhs.subs(c2, c2_coeffs)
print("Result")
print(solved_diffeq_rhs)
print()
```
Output:
```python
Original
(a0*(-uc + u(t))*(-urest + u(t)) + i*r)/tau
Simplified
c0 + c1*u(t) + c2*u(t)**2
Full equation
Eq(Derivative(u(t), t), c0 + c1*u(t) + c2*u(t)**2)
Result
-t + sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*log(uc - 2*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(a0*uc*urest/tau + i*r/tau) + tau*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(-a0*uc/tau - a0*urest/tau)**2/(2*a0) + tau*(-a0*uc/tau - a0*urest/tau)/(2*a0)) - sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*log(uc + 2*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(a0*uc*urest/tau + i*r/tau) - tau*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(-a0*uc/tau - a0*urest/tau)**2/(2*a0) + tau*(-a0*uc/tau - a0*urest/tau)/(2*a0))
```
## [Numerical Evaluation](https://docs.sympy.org/latest/modules/evalf.html) ## [Numerical Evaluation](https://docs.sympy.org/latest/modules/evalf.html)
```python ```python
@ -307,3 +243,78 @@ import sympy
x = sympy.symbols("x") x = sympy.symbols("x")
sympy.solve([x**2 - 1, x >= 0.5, x <= 3], x) sympy.solve([x**2 - 1, x >= 0.5, x <= 3], x)
``` ```
## Example: Gain function of the quadratic integrate and fire neuron
```python
import sympy
import numpy as np
import matplotlib.pyplot as plt
tau_value: float = 10e-3 # s
a0_value: float = 0.1e3 # V^-1
uc_value: float = -55.0e-3 # V
urest_value: float = -70.0e-3 # V
r_value: float = 10e6 # Ohm
uthr_value: float = -40e-3 # V
u = sympy.symbols("u", cls=sympy.Function, real=True)
urest, uc, uthr = sympy.symbols("urest uc uthr", real=True)
t, tau, a0, r, i = sympy.symbols("t tau a0 r i", real=True, positive=True)
c0, c1, c2 = sympy.symbols("c0 c1 c2", real=True)
diffeq_rhs = (a0 * (u(t) - urest) * (u(t) - uc) + r * i) / tau
diffeq_rhs = sympy.expand(diffeq_rhs)
diffeq_rhs = sympy.collect(diffeq_rhs, u(t))
c0_coeffs = diffeq_rhs.coeff(u(t), 0)
c1_coeffs = diffeq_rhs.coeff(u(t), 1)
c2_coeffs = diffeq_rhs.coeff(u(t), 2)
diffeq_rhs = diffeq_rhs.subs(c0_coeffs, c0)
diffeq_rhs = diffeq_rhs.subs(c1_coeffs, c1)
diffeq_rhs = diffeq_rhs.subs(c2_coeffs, c2)
diffeq = sympy.Eq(u(t).diff(t), diffeq_rhs)
solved_diffeq = sympy.dsolve(diffeq, u(t), ics={u(0): urest}, simplify=False)
solved_diffeq = solved_diffeq.subs(u(t), uthr)
solved_diffeq = sympy.simplify(solved_diffeq)
solved2_diffeq = sympy.solve(solved_diffeq, t)[0]
solved2_diffeq = solved2_diffeq.subs(c0, c0_coeffs)
solved2_diffeq = solved2_diffeq.subs(c1, c1_coeffs)
solved2_diffeq = solved2_diffeq.subs(c2, c2_coeffs)
solved2_diffeq = sympy.simplify(solved2_diffeq)
solved2_diffeq = solved2_diffeq.subs(tau, tau_value)
solved2_diffeq = solved2_diffeq.subs(a0, a0_value)
solved2_diffeq = solved2_diffeq.subs(uc, uc_value)
solved2_diffeq = solved2_diffeq.subs(urest, urest_value)
solved2_diffeq = solved2_diffeq.subs(r, r_value)
solved2_diffeq = solved2_diffeq.subs(uthr, uthr_value)
solved2_diffeq = sympy.simplify(solved2_diffeq)
print("The final function")
print(solved2_diffeq)
print()
thr_func = sympy.lambdify(i, solved2_diffeq, "numpy")
i = 4 * 1e-9 * np.arange(0, 1000, dtype=np.complex128) / 1000
z = thr_func(i)
z = 1.0 / z
z[z < 0] = 0
z = z.astype(dtype=np.float32)
plt.plot(i * 1e9, z)
plt.xlabel("Input [nA]")
plt.ylabel("Rate [Hz]")
plt.show()
```