Slide 9

The Simple Model Exhibits Both a Spike Kink and Onset Threshold Variability in the Somatic Compartment

Figure 9.  Results from injection of noise into the somatic compartment of the simple model neuron.  A, D.  Spikes in the soma and axon initial segments.  Note the kink in the somatic spike at action potential onset.  B, E.  Phase plots of the action potentials in the somatic and axonal segments.  Note the biphasic nature of spike generation in the soma and the appearance of a "kink".  C, F.  Variation in the spike onset voltage in the soma and axon initial segment.  Note the increased variability in the soma.

The noise that we injected is available here.

The model runs in NEURON and the model files can be obtained here.

Below are the equations and parameters used in this simple model of neuronal action potential generation.

Currents:

Only spiking related fast Na and K channels are included.

We use a Hodgkin and Huxley style model

C * dv/dt = Iext - INa  - IKv - Ileak

Iion = g.a^x .b(v - E) where g is the local conductance density; a is an activation variable with x order kinetics, b is an optional inactivation variable; v is the local membrane potential, and E is the reversal potential for the ionic species

(Eleak = -70 mV, EK = -90 mV, ENa = 60 mV).

Equations for Na+ current:

INa = gNa * m^3 *h *(v-ENa)
m_inf = alpha_m/(alpha_m+beta_m)
m_tau = 1/(alpha_m +beta_m)
dm/dt = alpha*(1-m)-beta*m
or
m_tau*dm/dt = -m +m_inf
alpha = 0.182(v+30)/(1-exp(-(v+30)/9))
beta = -0.124(v+30)/(1-exp((v+30)/9))

h_inf = 1/(1+exp(exp(v+60)/6.2)
h_tau = 1/((alpha_h +beta_h)
dh/dt = alpha*(1-h)-beta*h
or
h_tau*dh/dt = -h +h_inf
alpha = 0.024(v+45)/(1-exp(-(v+45)/5))
beta = -0.0091(v+70)/(1-exp((v+70)/5))

Equations for K+ current:

Ikv= gKv * n *(v-EKv)

n_inf = alpha_m/(alpha_m+beta_m)
n_tau = 1/(alpha_m +beta_m)
dn/dt = alpha*(1-n)-beta*n
or
n_tau*dn/dt = -n +n_inf
alpha = 0.004(v-30)/(1-exp(-(v-30)/9))
beta = -0.0004(v-30)/(1-exp((v-30)/9))

The temperature of the Simulation was 37o C:
Celsius=37;
q10=2.3;
temp=23;
tadj= q10^((Celsius-temp)/10);
[I_ion = tadj* g_ion*a^x*b*(v-E)]


Background Leak Current:

Ileak = gl (v-El)

The equations can be obtained from figure 1 legend in the paper by Mainen ZF, Sejnowski TJ. Influence of dendritic structure on firing pattern in model neocortical  neurons. Nature. 1996 Jul 25;382(6589):363-6.  The K+ current was revised to match the action potential behavior or layer 5 pyramidal cells in vitro.

Structure of the Simple Model:

The dendrite consisted of 300 compartments and was 3000 microns long and 8 microns wide.  The soma consisted of 5 compartments and measured 20 microns wide and 30 microns long.  The axon hillock was modeled as 2 compartments. Compartment 1 was 2 microns wide and 5 microns long, while the second was 1 microns wide and 5 microns long.  Finally, the axon initial segment  consisted of 8 compartments, length 40 microns, and 1 micron diameter.

The distribution of gNa+ and gK+ was as follows:

gNa+: dendrite: 100 ; soma: 500; hillock and axon IS: 7500
gK+: dendrite: 2; Soma: 320; hillock and axon IS: 1600
All values are in picosiemens per micron2.
Cm=0.75 uF/cm^2
Rm = 30,000 ohm/cm^2
axial resistance: 100 ohm-cm