The cable equation

Chapter 6.2

Read chapter 6.2

Python / NEURON demonstration

Imports:

from neuron import h, gui
import numpy as np
import matplotlib.pyplot as plt

Model definition (stick and ball):

soma = h.Section(name='soma') 
soma.L = soma.diam = 12.6157 # [um]
soma.Ra = 100                # Axial resistance [Ohm * cm]
soma.cm = 1                  # Membrane capacitance [uF / cm^2]
soma.insert('hh')            # Insert active Hodgkin-Huxley current in the soma
soma.gnabar_hh = 0.12        # Sodium conductance [S/cm2]
soma.gkbar_hh = 0.036        # Potassium conductance [S/cm2]
soma.gl_hh = 0.0003          # Leak conductance [S/cm2]
soma.el_hh = -54.3           # Reversal potential [mV]

dend = h.Section(name='dend')
dend.L = 200       # [um]
dend.nseg = 101
dend.Ra = 100      # Axial resistance [Ohm * cm]
dend.cm = 1        # Membrane capacitance [uF / cm^2]
dend.diam = 1      # [um]
dend.insert('pas') # Insert passive current in the dendrite
dend.g_pas = 0.001 # Passive conductance [S/cm2] 
dend.e_pas = -65   # Leak reversal potential [mV]
dend.connect(soma(1)) 

Stimulation:

stim = h.IClamp(dend(1))
stim.delay = 5
stim.dur = 1
stim_amp_array = [0.1, 0.3]

Recording vectors:

t_vec = h.Vector() 
t_vec.record(h._ref_t)

soma_v_vec = h.Vector()  
soma_v_vec.record(soma(0.5)._ref_v)

dend_v_vec_array = []
string_array     = []

for i in np.flip(np.linspace(0, dend.L, 6)):
    dend_v_vec = h.Vector()
    string_array.append('{} %'.format(100*(i/dend.L)))
    dend_v_vec.record(dend(i/dend.L)._ref_v)
    dend_v_vec_array.append(dend_v_vec)

Simulation parameters:

simdur = 25.0
h.tstop = simdur

Simulating:

for s in stim_amp_array:
    stim.amp = s
    h.run()

Plotting:

    plt.figure(figsize=(10,5))
    cmap = plt.get_cmap('Blues')   
    colors = cmap(np.linspace(0,1,len(dend_v_vec_array) * 2))
    plt.figure(figsize=(8,4)) 

    plt.plot(t_vec, dend_v_vec_array[0], label='dendrite @ {}'.format(string_array[0]), linewidth = 3, color = colors[-1]) 
    for i,v in enumerate(dend_v_vec_array[1:]):
        plt.plot(t_vec, v, label='dendrite @ {}'.format(string_array[i+1]), color = colors[len(colors)-2-i]) 
    plt.plot(t_vec, soma_v_vec, label='soma', color = 'red', linewidth=3)
    plt.title("Cable Equation", fontSize=15)
    plt.xlim([5,11])
    plt.xlabel('Time (ms)', fontSize=15) 
    plt.ylabel("Membrane Potential (mV)", fontSize=15) 
    plt.legend()
    plt.show()
    
h("forall {delete_section()}")

Resulted below threshold dynamic:

Resulted above threshold dynamic:

Modeling for various resulutions:

from neuron import h, gui
import numpy as np
import matplotlib.pyplot as plt

soma = h.Section(name='soma') 
soma.L = soma.diam = 12.6157 # [um]
soma.Ra = 100                # Axial resistance [Ohm * cm]
soma.cm = 1                  # Membrane capacitance [uF / cm^2]
soma.insert('hh')            # Insert active Hodgkin-Huxley current in the soma
soma.gnabar_hh = 0.12        # Sodium conductance [S/cm2]
soma.gkbar_hh = 0.036        # Potassium conductance [S/cm2]
soma.gl_hh = 0.0003          # Leak conductance [S/cm2]
soma.el_hh = -54.3           # Reversal potential [mV]

dend = h.Section(name='dend')
dend.L = 200       # [um]
dend.Ra = 100      # Axial resistance [Ohm * cm]
dend.cm = 1        # Membrane capacitance [uF / cm^2]
dend.diam = 1      # [um]
dend.insert('pas') # Insert passive current in the dendrite
dend.g_pas = 0.001 # Passive conductance [S/cm2] 
dend.e_pas = -65   # Leak reversal potential [mV]
dend.connect(soma(1)) 

stim = h.IClamp(dend(1))
stim.delay = 5
stim.dur = 1
stim.amp = 0.3

resolution_array = [2 , 4, 10]

t_vec = h.Vector() 
t_vec.record(h._ref_t)

soma_v_vec = h.Vector()  
soma_v_vec.record(soma(0.5)._ref_v)

dend_v_vec = h.Vector()
dend_v_vec.record(dend(1)._ref_v)
    
simdur = 25.0
h.tstop = simdur 

plt.figure(figsize=(10,5))
line_types = [':', '--', '-']
for i, r in enumerate(resolution_array):

    dend.nseg = r

    h.run()
    plt.plot(t_vec, dend_v_vec, label='dendrite with {} partitions'.format(r), color = 'black', linewidth = 3, linestyle=line_types[i]) 
    plt.plot(t_vec, soma_v_vec, label='soma', color = 'red', linewidth=3, linestyle=line_types[i])
    
plt.title("Cable Equation", fontSize=15)
plt.xlim([5,11])
plt.xlabel('Time (ms)', fontSize=15) 
plt.ylabel("Membrane Potential (mV)", fontSize=15) 
plt.legend()
plt.show()
    
h("forall {delete_section()}")

Results: