neurotools.obsolete.models.lif module

Leaky integrate-and-fire model

neurotools.obsolete.models.lif.euler_integrate_LIF(x, C=1.0, g_L=0.1, g_E=0.00074, E_E=0.0, E_L=-70.0, Thr=-60.0, tau=1.0, Fs=2000, g_S=None)

Modeled after the LIF implementation in Baker Pinches Lemon 2003 This one uses Euler integration though – it appears similar

C dV/dt = g_L (E_L - V) + g_E (E_E - V)

x is synaptic input in units of “counts” for synaptic activation events. C = 1 uF/cm^2 membrane capacitance g_L = 0.1 mS/cm^2 leak conductance E_L = -70 mV resting potential g_E = 0.00074 mS/cm^2 excitatory conductance E_E = 0 mV Thr = -60 mV spiking threshold tau = 1.0 ms synaptic time constant in seconds Fs = 2000 Hz sampling rate and integration frequency V = mV membrane potential

Note: Expected membrane time constant is 10ms Tau = RC = C/g_L = 1 uF / 0.1 mS = 10 ms. Checks out. Note: Expected EPSP is 100 microvolts (0.1mV) Emperically this checks out

Unit check. Farad Volt / Second = Ampere = Siemen * Volt = Volt / Ohm uF * mV / ms = mS * mV 1 millisiemen * 1 millivolt = 1 microamp 1 microamp / 1 microfarad * 1 millisecond = 1 millivolt

# Test code: single EPSP x = zeros(2000) x[100] = 1 time, V, spikes, current = euler_integrate_LIF(x) clf() plot(time,V) plot(time,current) plot(time,x)

neurotools.obsolete.models.lif.exponential_integrate_LIF(x, C=1.0, g_L=0.1, g_E=0.00074, E_E=0.0, E_L=-70.0, Thr=-60.0, tau=1.0, Fs=2000.0, g_S=None)

Modeled after the LIF implementation in Baker Pinches Lemon 2003

C dV/dt = g_L (E_L - V) + g_E (E_E - V)

x is synaptic input in units of “counts” for synaptic activation events. C = 1 uF/cm^2 membrane capacitance g_L = 0.1 mS/cm^2 leak conductance E_L = -70 mV resting potential g_E = 0.00074 mS/cm^2 excitatory conductance E_E = 0 mV Thr = -60 mV spiking threshold tau = 1.0 ms synaptic time constant in seconds Fs = 2000 Hz sampling rate and integration frequency V = mV membrane potential

Note: Expected membrane time constant is 10ms Tau = RC = C/g_L = 1 uF / 0.1 mS = 10 ms. Checks out. Note: Expected EPSP is 100 microvolts (0.1mV) Emperically this checks out

Unit check. Farad Volt / Second = Ampere = Siemen * Volt = Volt / Ohm uF * mV / ms = mS * mV 1 millisiemen * 1 millivolt = 1 microamp 1 microamp / 1 microfarad * 1 millisecond = 1 millivolt

# Test code: single EPSP and check against Euler integrator x = zeros(2000) x[100] = 1 time, V, spikes, current = euler_integrate_LIF(x) clf() plot(time,V) plot(time,current) plot(time,x) time, V, spikes, current = exponential_integrate_LIF(x) plot(time,V) plot(time,current) plot(time,x) # they appear to match

neurotools.obsolete.models.lif.exponential_moving_average(x, tau, Fs=1000)

x : data tau : time constant in seconds Fs : sampling rate of x in samples per second

Implement exponential moving average as Y_{n+1} = (1-alpha) Y_n + alpha X_n

This relates to convolving signal x with decaying exponential Y = X * [H(t) exp(-t/tau)] Where t is in seconds and H is the heaviside step function

Alpha and tau may be related by considering the differential equation

tau dY/dT = X-Y

And both solving it as a linear equation and also re-writing it as a discrete difference equation

DY = (X-Y) DT/tau [Y_{n+1}-Y] = (X-Y) DT/tau Y_{n+1} = (X-Y) DT/tau + Y Y_{n+1} = (1 - DT/tau) Y + DT/tau X

and we find that alpha = DT/tau for the discrete update

The exact solution to an impulse in X would be

tau dY/dT = -Y, Y_0 = 1

Y(t) = exp(-t/tau) * H(t) or Y(t) = H(t) exp(-t*alpha/DT)