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cb_models.py
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cb_models.py
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from abc import ABC, abstractmethod
from scipy.integrate import solve_ivp
import numpy as np
from numpy import exp
from copy import deepcopy
class HHKinetics(ABC):
"""
HH-type (alpha-beta) kinetics abstract class.
Has to be implemented by an activation or inactivation subclass.
"""
@abstractmethod
def alpha(self,V):
pass
@abstractmethod
def beta(self,V):
pass
def inf(self,V):
return self.alpha(V) / (self.alpha(V) + self.beta(V))
def tau(self, V):
return 1 / (self.alpha(V) + self.beta(V))
def vfield(self, x, V, Vpost = None):
if Vpost is None:
# Intrinsic kinetics vector field
return self.alpha(V) * (1 - x) - self.beta(V) * x
else:
# Synaptic kinetics vector field
return self.alpha(V) * (1 - x) - self.beta(Vpost) * x
class NeuroDynRate:
"""
NeuroDyn-type alpha or beta functions (kinetic rates)
"""
def __init__(self, Ib, kappa, Vt, Vb, sign):
self.Ib = Ib
self.perturbations = np.ones(np.size(Ib))
self.kappa = kappa
self.Vt = Vt
self.Vb = Vb
self.sign = sign
def I_rate(self, V):
I=0
Ib = self.Ib * self.perturbations
for i in range(np.size(self.Ib)):
I += Ib[i] / (1 + np.exp(self.sign * self.kappa * (self.Vb[i] - V) / self.Vt))
return I
def perturb(self, sigma = 0.15):
self.perturbations = (1 + sigma * np.random.randn(7))
class NeuroDynActivation(HHKinetics):
"""
NeuroDyn-type activation gating variable kinetics.
"""
def __init__(self, Ib, kappa, C, Vt, Vb):
self.C = C
self.Vt = Vt
self.alpharate = NeuroDynRate(Ib[0], kappa, Vt, Vb, 1)
self.betarate = NeuroDynRate(Ib[1], kappa, Vt, Vb, -1)
def alpha(self,V):
return self.alpharate.I_rate(V) / (self.C * self.Vt)
def beta(self,V):
return self.betarate.I_rate(V) / (self.C * self.Vt)
class NeuroDynInactivation(HHKinetics):
"""
NeuroDyn-type inactivation gating variable kinetics.
"""
def __init__(self, Ib, kappa, C, Vt, Vb):
self.C = C
self.Vt = Vt
self.alpharate = NeuroDynRate(Ib[0], kappa, Vt, Vb, -1)
self.betarate = NeuroDynRate(Ib[1], kappa, Vt, Vb, 1)
def alpha(self,V):
return self.alpharate.I_rate(V) / (self.C * self.Vt)
def beta(self,V):
return self.betarate.I_rate(V) / (self.C * self.Vt)
class HHActivation(HHKinetics):
"""
HH-type (alpha-beta) activation gating variable kinetics.
"""
def __init__(self, aVh, aA, aK, bVh, bA, bK, SI_units=False):
self.aVh = aVh
self.aA = aA
self.aK = aK
self.bVh = bVh
self.bA = bA
self.bK = bK
# Convert paramaters to SI units
if (SI_units):
self.aVh *= 1e-3
self.aA *= 1e6
self.aK *= 1e-3
self.bVh *= 1e-3
self.bA *= 1e3
self.bK *= 1e-3
def alpha(self, V):
A = self.aA
K = self.aK
Vh = self.aVh
V = np.asarray(V)
a = np.zeros(V.shape)
a[V != Vh] = A * (Vh - V[V != Vh]) / (exp((Vh - V[V != Vh]) / K) - 1)
a[V == Vh] = A * K
return a
def beta(self, V):
return self.bA * exp((self.bVh - V) / self.bK)
class HHInactivation(HHKinetics):
"""
HH-type (alpha-beta) inactivation gating variable kinetics.
"""
def __init__(self, aVh, aA, aK, bVh, bA, bK, SI_units=False):
self.aVh = aVh
self.aA = aA
self.aK = aK
self.bVh = bVh
self.bA = bA
self.bK = bK
# Convert paramaters to SI units
if (SI_units):
self.aVh *= 1e-3
self.aA *= 1e3
self.aK *= 1e-3
self.bVh *= 1e-3
self.bA *= 1e3
self.bK *= 1e-3
def alpha(self, V):
return self.aA * exp((self.aVh - V)/ self.aK)
def beta(self, V):
return self.bA / (exp((self.bVh - V) / self.bK) + 1)
class NeuronalModel(ABC):
"""
Abstract class for neuronal models implementing the simulate method.
"""
@abstractmethod
def vfield(self, x, I):
pass
def simulate(self, trange, x0, Iapp, mode="continuous"):
# Note: Iapp should be a function of t, e.g., Iapp = lambda t : I0
if mode == "continuous":
def odesys(t, x):
return self.vfield(x, Iapp(t))
return solve_ivp(odesys, trange, x0)
else:
#... code forward-Euler integration
return
def vclamp(self, trange, x0, V, mode = "continuous"):
# Note: V should be a function of t, e.g., V = lambda t : V0
if mode == "continuous":
def odesys(t, x):
return self.vclamp_vfield(x, V(t))
return solve_ivp(odesys, trange, x0)
else:
#... code forward-Euler integration
return
class NeuroDynModel(NeuronalModel):
"""
Model of a single NeuroDyn neuron.
"""
def __init__(self, dg = None, dErev = None, dIb = None, V_ref = 0.9,
I_voltage = 150e-9, I_master = 200e-9, I_ref = 100e-9,
capacitance_scaling = 1.0, digital_values = True):
self.digital_values = digital_values
# Number of states (needed for network class)
self.x_len = 4
if (dg is None):
dg = np.array([1023, 307, 3])
if (dErev is None):
dErev = np.array([829, -829, -545])
if (dIb is None):
dIb_m = np.array([[0, 1, 11, 23, 0, 0, 870],
[190, 4, 6, 0, 0, 0, 0]])
dIb_h = np.array([[3, 0, 0, 0, 0, 0, 0],
[0, 0, 6, 3, 0, 0, 0]])
dIb_n = np.array([[0, 0, 2, 2, 3, 0, 0],
[15, 0, 0, 0, 0, 0, 0]])
dIb = [dIb_m, dIb_h, dIb_n]
self.V_ref = V_ref # Unit V
self.I_voltage = I_voltage # Unit A
self.I_master = I_master # Unit A
self.I_ref = I_ref # Unit A
self.vHigh = self.V_ref + I_voltage*1.85*1e6
self.vLow = self.V_ref - I_voltage*1.85*1e6
# Membrane & gate capacitances
self.C_m = 4e-12 * capacitance_scaling # Unit F
self.C_gate = 5e-12 # Unit F
# Scaling parameters (e.g. parameters that set the voltage scale, time scale..)
self.kappa = 0.7
self.kappa_lin = 0.2 # linearized slope of conductance amplifiers
self.Vt = 26e-3 # Unit V
self.Res = 1.63e6 # Unit Ohm
# Digital parameters
self.dg = dg
self.dErev = dErev
self.dIb = dIb
# Perturbation arrays for g and Erev, initialize at 1
self.perturb_g = np.ones(dg.shape)
self.perturb_Erev = np.ones(dErev.shape)
# Convert digital to physical values
self.gna, self.gk, self.gl = self.convert_conductance(dg)
self.Ena, self.Ek, self.El = self.convert_potential(dErev)
Ib_m = self.convert_current(dIb[0])
Ib_h = self.convert_current(dIb[1])
Ib_n = self.convert_current(dIb[2])
# Gating variable coefficients
self.p = 3
self.q = 1
self.r = 4
Vb = self.get_Vb()
self.Vb = Vb
# Construct gating variables
self.m = NeuroDynActivation(Ib_m, self.kappa, self.C_gate, self.Vt, Vb)
self.h = NeuroDynInactivation(Ib_h, self.kappa, self.C_gate, self.Vt, Vb)
self.n = NeuroDynActivation(Ib_n, self.kappa, self.C_gate, self.Vt, Vb)
def convert_current(self, dI):
dI = np.asarray(dI)
# Factor for converting digital to physical I
if (self.digital_values):
I_factor = self.I_master / 1024
else:
I_factor = 1
return dI * I_factor
def convert_conductance(self, dg):
dg = np.asarray(dg)
# Factor for converting digital to physical g
if (self.digital_values):
g_factor = (self.kappa_lin / self.Vt) * (self.I_master / 1024)
else:
g_factor = 1
return dg * g_factor
def convert_potential(self, dErev):
dErev = np.asarray(dErev)
# Factor for converting digital to physical Erev
if (self.digital_values):
E_factor = (self.I_voltage / 1024) * self.Res
else:
E_factor = 1
return dErev * E_factor + self.V_ref
def update_dg(self, dg):
dg = np.asarray(dg)
self.dg = dg
self.gna,self.gk,self.gl = self.convert_conductance(dg * self.perturb_g )
def update_dErev(self, dErev):
dErev = np.asarray(dErev)
self.dErev = dErev
self.Ena,self.Ek,self.El = self.convert_potential(self.dErev * self.perturb_Erev)
def update_dIb(self, dIb):
dIb = np.asarray(dIb)
self.dIb = dIb
Ib_m = self.convert_current(dIb[0])
Ib_h = self.convert_current(dIb[1])
Ib_n = self.convert_current(dIb[2])
Vb = self.Vb
self.m = NeuroDynActivation(Ib_m, self.kappa, self.C_gate, self.Vt, Vb)
self.h = NeuroDynInactivation(Ib_h, self.kappa, self.C_gate, self.Vt, Vb)
self.n = NeuroDynActivation(Ib_n, self.kappa, self.C_gate, self.Vt, Vb)
def get_pars(self):
params = {
'kappa': self.kappa,
'C': self.C_gate,
'C_ND': self.C_m,
'Vt': self.Vt,
'Res': self.Res
}
return params
def get_Vb(self):
# Bias voltages for the 7-point spline regression
Vb = np.zeros(7) # Define the 7 bias voltages
I_factor = (self.vHigh - self.vLow) / 700e-3
Vb[0] = self.vLow + (I_factor * 50e-3)
for i in range(1, 7):
Vb[i] = Vb[i-1] + (I_factor * 100e-3)
return Vb
def resistor(self, g, V, linear=False):
if (linear):
I = g * V
else:
k = self.kappa_lin
Vt = self.Vt
I = 2 * g * Vt /k * np.tanh(k * V / (2 * Vt))
return I
def Ina(self, V, m, h):
return self.resistor(self.gna * (m**self.p) * (h**self.q), V - self.Ena)
def Ik(self, V, n):
return self.resistor(self.gk * (n**self.r), V - self.Ek)
def Il(self, V):
return self.resistor(self.gl, V - self.El)
def i_int(self,V, m, h, n):
return (self.Ina(V, m , h) + self.Ik(V, n) + self.Il(V))
def vfield(self, x, I):
V, m, h, n = x
dV = (-self.i_int(V, m, h, n) + I) / self.C_m
dm = self.m.vfield(m, V)
dh = self.h.vfield(h, V)
dn = self.n.vfield(n, V)
return [dV, dm, dh, dn]
def vclamp_vfield(self, x, V):
m, h, n = x
dm = self.m.vfield(m, V)
dh = self.h.vfield(h, V)
dn = self.n.vfield(n, V)
return [dm, dh, dn]
def perturb(self, sigma = 0.15):
# Pertrub exponents
self.p = 3 + 0.2*np.random.randn()
self.q = 1 + 0.1*np.random.randn()
self.r = 4 + 0.2*np.random.randn()
# For each alpha/beta, perturb sigmoid base currents
for x in [self.m, self.h, self.n]:
x.alpharate.perturb(sigma)
x.betarate.perturb(sigma)
# Update perturbation arrays for g and Erev
self.perturb_g = 1 + sigma * np.random.randn(*self.dg.shape)
self.perturb_Erev = 1 + sigma * np.random.randn(*self.dErev.shape)
# Update g and Erev
self.gna,self.gk,self.gl = self.convert_conductance(self.dg * self.perturb_g)
self.Ena,self.Ek,self.El = self.convert_potential(self.dErev * self.perturb_Erev)
# Perturb voltage offsets?
# Would add ~15mV sigma to each 'bias' voltage
class HHModel(NeuronalModel):
"""
Hodgkin-Huxley model of a single neuron.
"""
def __init__(self, gna = 120, gk = 36, gl = 0.3, Ena = 120, Ek = -12,
El = 10.6, gates = None, scl_v = 1, scl_t = 1, SI_units = False):
# Number of states (needed for network class)
self.x_len = 4
self.C_m = 1
self.gna = gna*scl_t
self.gk = gk*scl_t
self.gl = gl*scl_t
self.Ena = Ena*scl_v
self.Ek = Ek*scl_v
self.El = El*scl_v
self.scl_v = scl_v
self.scl_t = scl_t
self.SI_units = SI_units
# Convert to SI units
if (SI_units):
self.C_m *= 1e-6
self.gna *= 1e-3
self.gk *= 1e-3
self.gl *= 1e-3
self.Ena *= 1e-3
self.Ek *= 1e-3
self.El *= 1e-3
if gates is None:
# Default to nominal HH kinetics
self.m = HHActivation(25*scl_v, 0.1*scl_t/scl_v, 10*scl_v, 0*scl_v,
4*scl_t, 18*scl_v, SI_units)
self.h = HHInactivation(0*scl_v, 0.07*scl_t, 20*scl_v, 30*scl_v,
1*scl_t, 10*scl_v, SI_units)
self.n = HHActivation(10*scl_v, 0.01*scl_t/scl_v, 10*scl_v,
0*scl_v, 0.125*scl_t, 80*scl_v, SI_units)
else:
# Note: gates should be passed scaled if scl_v/scl_t != 1
self.m = gates[0]
self.h = gates[1]
self.n = gates[2]
# Gating variable coefficients
self.p = 3
self.q = 1
self.r = 4
# Save the nominal parameters
self.nominal = deepcopy(self)
def Ina(self, V, m, h):
return self.gna * (m**self.p) * (h**self.q) * (V - self.Ena)
def Ik(self, V, n):
return self.gk * (n**self.r) * (V - self.Ek)
def Il(self, V):
return self.gl * (V - self.El)
def i_int(self, V, m, h, n):
return self.Ina(V, m, h) + self.Ik(V, n) + self.Il(V)
def vfield(self, x, I):
V, m, h, n = x
# Scale external current
I = I * self.scl_v * self.scl_t
dV = (-self.i_int(V, m, h, n) + I) / self.C_m
dm = self.m.vfield(m, V)
dh = self.h.vfield(h, V)
dn = self.n.vfield(n, V)
return [dV, dm, dh, dn]
def vclamp_vfield(self, x, V):
m, h, n = x
dm = self.m.vfield(m, V)
dh = self.h.vfield(h, V)
dn = self.n.vfield(n, V)
return [dm, dh, dn]
def perturb(self, sigma = 0.15):
nom = self.nominal
# Pertrub exponents
self.p = nom.p + 0.2 * np.random.randn()
self.q = nom.q + 0.1 * np.random.randn()
self.r = nom.r + 0.2 * np.random.randn()
# Perturb maximal conductances
self.gna = nom.gna * (1 + sigma * np.random.randn())
self.gk = nom.gk * (1 + sigma * np.random.randn())
self.gl = nom.gl * (1 + sigma * np.random.randn())
# Perturb reversal potential
self.Ena = nom.Ena * (1 + sigma * np.random.randn())
self.Ek = nom.Ek * (1 + sigma * np.random.randn())
self.El = nom.El * (1 + sigma * np.random.randn())
# Perturb alpha/beta rates
gates = [self.m, self.h, self.n]
nom_gates = [nom.m, nom.h, nom.n]
for x, x_nom in zip(gates, nom_gates):
x.aA = x_nom.aA * (1 + sigma*np.random.randn())
x.bA = x_nom.bA * (1 + sigma*np.random.randn())
class ShortCircuit(NeuronalModel):
"""
Model defined as a short-circuit of several Hodgkin-Huxley or NeuroDyn
models
Arguments:
neurons: list of neurons to be short-circuited
"""
def __init__(self, neurons):
self.neurons = neurons
# Number of states
x_lens = [neuron.x_len for neuron in neurons]
self.x_len = np.sum(x_lens) - len(neurons) + 1
# Find total capacitance
self.C_m = 0
for j, neuron in enumerate(neurons):
self.C_m += neuron.C_m
def vfield(self, x, I):
V = x[0]
i_int = 0
dx = [0]
for j, neuron in enumerate(self.neurons):
m = x[1+j*3]
h = x[2+j*3]
n = x[3+j*3]
i_int += neuron.i_int(V, m, h, n)
dx_j = neuron.vfield([V, m, h, n], I)
dx.extend(dx_j[1:4])
dV = (-i_int + I) / self.C_m
dx[0] = dV
return dx
##### NETWORK-RELATED CLASSES #####
class Synapse:
"""
Arbitrary synapse class
Arguments:
gsyn: maximal conductance
Esyn: synapse reversal potential
r: synapse activation kinetics
"""
def __init__(self, gsyn, Esyn, r):
self.gsyn = gsyn
self.Esyn = Esyn
self.r = r # HHKinetics class
def Iout(self, r, Vpost):
return self.gsyn * r * (Vpost - self.Esyn)
class AMPA(HHKinetics):
"""
AMPA gating variable kinetics
Physiological values taken from Ermentrout et al. 2010, p. 161
Note: Voltage values need to be shifted +65mV
"""
def __init__(self, Tmax = 1, Kp = 5, V_T = 2+65, ar = 1.1, ad = 0.19,
scl_v = 1, SI_units = False):
self.Tmax = Tmax
self.Kp = Kp * scl_v
self.V_T = V_T * scl_v
self.ar = ar
self.ad = ad
if (SI_units):
self.ar *= 1e3
self.ad *= 1e3
self.Kp *= 1e-3
self.V_T *= 1e-3
def alpha(self, V):
return self.ar * self.Tmax / (1 + np.exp(-(V - self.V_T) / self.Kp))
def beta(self, V):
V = np.asarray(V)
return self.ad * np.ones(V.shape)
class AMPASynapse(Synapse):
"""
AMPA synapse with parameters taken from Ermentrout et al. 2010, p. 161
"""
def __init__(self, gsyn):
super().__init__(gsyn, 65, AMPA())
class NDSynapse(Synapse):
def __init__(self, dg = 0, dE = 130, dIb = None, ND = None):
# Set parent NeuroDyn chip to get the parameters
if (ND is None):
ND = NeuroDynModel()
self.ND = ND
if dIb is None:
dIb = np.array([[0, 0, 0, 0, 0, 0, 378],
[0, 0, 0, 0, 0, 0, 1]])
# Digital values
self.dg = dg
self.dE = dE
self.dIb = dIb
# Convert to analog
g = ND.convert_conductance(dg)
E = ND.convert_potential(dE)
Ib = ND.convert_current(dIb)
# Define the activation variable
r = NeuroDynActivation(Ib, ND.kappa, ND.C_gate, ND.Vt, ND.Vb)
# Initialize synapse parameters
super().__init__(g, E, r)
def update_dg(self, dg):
self.dg = dg
self.gsyn = self.ND.convert_conductance(dg)
def update_dE(self, dE):
self.dE = dE
self.Esyn = self.ND.convert_potential(dE)
def update_dIb(self, dIb):
dIb = np.asarray(dIb)
self.dIb = dIb
ND = self.ND
Ib = ND.convert_current(dIb)
self.r = NeuroDynActivation(Ib, ND.kappa, ND.C_gate, ND.Vt, ND.Vb)
class NeuronalNetwork(NeuronalModel):
"""
Neuronal network class (biophysical or neuromorphic)
Arguments:
gap : a gap junction adjacency matrix containing conductance values
syns : a matrix defining the synaptic connections:
- syns[i][j] == None: no synaptic connection from neuron j to
neuron i
- otherwise, syns[i][j] is a list of synapse objects from neuron j
to i
"""
def __init__(self, neurons, gap = None, syns = None):
self.neurons = neurons
self.gap = gap
self.syns = syns
def vfield(self, x, I):
dx = []
dx_syn = []
# State size for each neuron
x_lens = [neuron.x_len for neuron in self.neurons]
# Index of the first synapse state
idx_syn = sum(x_lens)
# i_x[i] is the index of the first state for neuron i
i_x = np.cumsum(np.pad(x_lens[:-1], (1, 0)))
for i, neuron_i in enumerate(self.neurons):
# Total synaptic and gap junction current to neuron i
i_syn = 0
i_gap = 0
Vpost = x[i_x[i]]
for j, _ in enumerate(self.neurons):
Vpre = x[i_x[j]]
# Go through all synaptic connections j -> i
if (self.syns is not None):
if (self.syns[i][j] is not None):
for syn in self.syns[i][j]:
r = x[idx_syn] # activation of the synapse
i_syn += syn.Iout(r, Vpost)
dx_syn.append(syn.r.vfield(r, Vpre, Vpost))
idx_syn += 1
if (self.gap is not None):
i_gap += self.gap[i][j] * (Vpost - Vpre)
# Total current to neuron i
Iext = I[i] - i_syn - i_gap
# Start and end indices for states of neuron i
i_start = i_x[i]
i_end = i_x[i] + x_lens[i]
# Add dx for neuron i
dx.extend(neuron_i.vfield(x[i_start:i_end], Iext))
dx.extend(dx_syn)
return dx
class NeuroDynBoard(NeuronalNetwork):
"""
Network class representing a single NeuroDyn board consisting of four
NeuroDyn neurons with full synaptic connectivity between them. Additional
resistive (gap junction) connections can also be defined.
"""
def __init__(self):
# Define neuronal models
neurons = [NeuroDynModel() for i in range(4)]
# Define synapses
syns = [[[NDSynapse(ND = neurons[0])] if (i != j) else None for j in range (4)] for i in range (4)]
# Set gap junction matrix to zeros
gap = np.zeros((4, 4))
super().__init__(neurons, syns = syns, gap = gap)
def get_neuron(self, i):
return self.neurons[i]
def get_syn(self, i, j):
return self.syns[i][j][0]
def set_gap(self, g, i, j):
self.gap[i][j] = g
self.gap[j][i] = g
class NeuroCube(NeuronalNetwork):
"""
Network class for a parallel interconnection of four NeuroDyn boards.
"""
def __init__(self):
boards = [NeuroDynBoard() for i in range(4)]
self.boards = boards
# Group all neurons
neurons = [neuron for board in boards for neuron in board.neurons]
# Group all synapses
syns = np.array([[None for j in range(16)] for i in range(16)])
for i in range(4):
syns[i*4:(i+1)*4, i*4:(i+1)*4] = boards[i].syns
# Group all gap junctions
gap = np.zeros((16, 16))
for i in range(4):
gap[i*4:(i+1)*4, i*4:(i+1)*4] = boards[i].gap
# Now repeat backward so that gap matrix of each board references
# to the accumulated gap matrix
for i in range(4):
boards[i].gap = gap[i*4:(i+1)*4, i*4:(i+1)*4]
super().__init__(neurons, syns = syns, gap = gap)
def connect_boards(self, board_i, board_j, neuron_no, g):
"""
Set the conductance to g of the connection between boards indexed by
board_i and board_j, through the neuron indexed neuron_no.
"""
neuron_i = board_i*4 + neuron_no
neuron_j = board_j*4 + neuron_no
self.gap[neuron_i][neuron_j] = g
self.gap[neuron_j][neuron_i] = g
def get_board(self, i):
return self.boards[i]