Introduction to Cognitive Science for Mathematical Scientists

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Notes on Neuron Dynamics: The Action Potential

Neural Currents I: Analysis Summary

September 5, 2003

	Equation	Quantity	Prerequisite
a. b.		Hodgkin-Huxley Eqn: Neural currents I Voltage-dependent current gates i Cable Equation	I = gV Ohm’s Law g = g1 + g2 Parallel conductances I = CdV/dt Capacitance Ei Resting potentials (c) g(V, t) Gates: H-H model (f)
c.	Ei = - (kT/ze) ln([Xi]out/[Xi]in)	Nernst Equation Resting potentials Ei	Nernst-Planck Equation (d)
d.		Nernst-Planck Equation Currents from drift, diffusion	Jdrift , Jdiff equations
f.		Gate conductance as function of gating particle states	Hodgkin-Huxley gate model
g.	sim. for m(t), h(t)	Time-dependence of gates	Hodgkin-Huxley gate model: reaction kinetics
h.		Voltage-dependence of gates	Energy barrier model Boltzmann distribution

Key:

q charge (coulombs)

I current (charge/sec= amperes)

J current density (charge/sec-cm²)

E electric field (force/charge)

V voltage = potential energy/charge (joules/coulomb) E = dV/dt

R resistance (to current flow; ohms Ω V = IR

g conductance = 1/R J = sE

s conductivity (inherent medium conductance) g_{1+2 in parallel} = g₁ + g₂

C capacitance I = C dV/dt

Introduction to Cognitive Science for Mathematical Scientists

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Neural Currents II: Fundamentals of Electric Currents

September 5, 2003

(1) Force, energy

a. Kinetic Energy K º ½mv² — [kg-m²/s² º joule] because:

b. dK/dt = mv dv/dt = v ma = vF — by Newton’s Second Law of Motion …

c. dK/dt = F dx/dt

d. DK = F Dx

e. Potential energy (work) U º - F Dx [actually, U º - ∫ F × dx, ‘ × ‘ = vector inner product]

f. Total energy ℰ º K + U — because:

g. Dℰ = 0 — Conservation of Total Energy

h. At temperature T (in °K, degrees Kelvin; k º Boltzmann’s constant):

Scale of molecular energy = kT — k = 1.38 × 10^-23 joule/°K

(2) Momentum

a. F = ma = m dv/dt = d (mv)/dt = dp/dt — [kg-m/s² º newton] where

b. p º mv is momentum

c. Vector quantity: p = mv — i.e., (p₁, p₂, p₃) = m(v₁, v₂, v₃) = (mv₁, mv₂, mv₃)

d. dp/dt = F — i.e., (dp₁/dt, …) = (F₁, …)

e. Dp = F Dt ‘impulse’

f. Conserved: dp_universe/dt = 0 — why?

g. F_{on 1 from 2} = -F_{on 2 from 1} — Newton’s First Law of Motion …

h. d(p₁+p₂)/dt = F_{on 1 from 2} + F_{on 2 from 1} = 0 — from d

(3) Electric charge, current, force, fields, potential

a. Electric charge, q (for electron, e)

b. Coulomb’s law: — [k = 1 iff q in coulombs]

c. In general: F = qE [E in volts/m]

d. Potential Energy difference º DU

= F Dx = qE Dx

e. Potential Energy difference per unit charge

º voltage drop º V º DU/q = E Dx [V in joules/coulomb º volts]

f. Current charge/sec: I or I [coulombs/s º amperes]

g. Conservation of charge: Σ_i I_i = 0 — Kirchoff’s Law ({I_i} = the currents into a point)

h. Current density, current/cm²: J or J

(4) Resistance (R), conductance (g)

a. V = IR I = gV (g = 1/R) — [R in ohms, Ω] Ohm’s Law. “Resistor”. Why?

b. Current density J; I º JA

c. Mysterious fact: J = σE — σ = conductivity

d. V = Dx E = Dx J/σ = Dx I/Aσ º IR; R º Dx/Aσ

e. Why the mysterious fact? — model of electrical conductance

i. key: scattering of e^- off lattice

ii. e^- collides with molecule on average after time τ (giving momentum to the lattice)

iii. hard sphere collision: outgoing direction of e^- is random

iv. let v be the velocity of the e^- coming out of last collision, time t ago

v. during time t gains momentum F t = eE t

vi. momentum at end of time t is p = mv + eEt

vii. average momentum of all electrons is ápñ = mávñ + eEátñ = 0 + eEτ

viii. this is m(effective [drift] velocity of e^- cloud) º mv̅

ix. v̅ = (eτ/m)E (cm/sec)

x. suppose density of e^-s is n per cm³

xi. J º current density º (charge flowing through 1 cm² in 1 sec) = (charge in a volume v cm³)

= nev̅ = ne²τ/m E

xii. J = sE — s º ne²τ/m

xiii. Generalizes to other charge carriers and non-hard-sphere-collisions (τ º randomization time)

(5) Resistors in the combinatorial strategy

a. Wires: R = 0, V = 0

b. Resistors in series

i. V adds, I in common, so …

ii. R₁₂ = R₁ + R₂

iii. Check: (4)d says R º Dx/Aσ ; Dx₁₂ = Dx₁ + Dx₂ ; A, s unchanged

c. Resistors in parallel

i. I adds, V in common, so …

ii. g₁₂ = g₁ + g₂

iii. Check: (4)d says R º Dx/Aσ ; A₁₂ = A₁ + A₂ ; Dx, s unchanged

(6) Capacitance

a. Charge can accumulate on/in materials

b. E.g.: parallel conducting plates; “capacitor”

c. Q = C V — due to relation between Q and E, Coulomb’s law

d. I = dQ/dt = C dV/dt — I in to capacitor

e. The basic RC circuit:

i. As current flows through a medium with a given R, C, equivalent to a parallel circuit

ii. V/R = I = -C dV/dt —‘-’ because here I is out from capacitor

iii. dV/dt = -(1/τ)V ; τ º RC

iv. V = V₀ e ^—t/τ — τ = time constant of exponential decay, V₀ º V(t=0)

Introduction to Cognitive Science for Mathematical Scientists

Cognitive Science 050.313/613

Neural Currents III: Qualitative Analysis

September 5, 2003

(1) Background

a. Ions most important in biological currents: K⁺, Na⁺, Cl^-, Ca⁺⁺ ; inside cell, other anions A^-

Typical mammalian cell:

Ca⁺⁺ Cl^- Na⁺ Na⁺ Cl^- Ca⁺⁺

A^- K⁺ K⁺ A^-

b. Space-charge neutrality normally holds (total +q) = (total -q)

(2) Ion concentrations

a. Are constant, even during signal propagation, to within .01%.

b. Primary determinants:

i. Membrane is impermeable to anions (organic acids and proteins) A^- concentrated inside

ii. Membrane is highly permeable to K⁺

iii. Membrane is moderately permeable to Cl^-

iv. Axon: At resting voltages (~ -70 mV), membrane is very slightly permeable to Na+;
as voltage increases (above ~ -50 mV), membrane becomes increasingly permeable to Na⁺,
becoming extremely permeable (high relative to K⁺) at peak of action potential (variation in permeability: g_Na from .05g_K to 500g_K.

c. Na-K pump

i. 3Na⁺ out for every 2K+ in

ii. not a major factor in determining most resting concentrations, or short-term behavior, but …

iii. essential over long term for enabling unequal concentrations to be maintained after action potentials, which involve Na⁺, K⁺ currents that would eventually alter resting concentrations

(3) Consequences at rest

a. Start with A^-, K⁺ and Cl^- concentrations

i. Space-charge neutrality outside Þ [Cl^-]out = [K⁺]out

ii. Space-charge neutrality inside (given A^- inside) Þ [Cl^-]_in < [K⁺]_in

b. Can’t have [K⁺]_in < [K⁺]_out

i. For contradiction, assume so. Then have [Cl^-]_in < [Cl^-]_out Þ Cl^- diffusion current inward

ii. therefore equilibrium Þ Cl^- drift current outward Þ

iii. E directed inward Þ

iv. K⁺ drift current inward; but if [K⁺]_in < [K⁺]_out then K⁺ diffusion is inward too: no equilibrium

c. Therefore must have [K⁺]_in > [K⁺]_out and …

i. must have V º V_in < V_out º 0,

net -charge on inside of membrane, net +charge on outside: no overall charge in/outside and …

ii. must have [Cl^-]_out > [Cl^-]_in .

d. Now add Na⁺:

i. Low membrane permeability to Na+ means low Na+ current, just to cancel pump Na+ current

ii. Pump sends Na⁺ outward (and K⁺ inward, opposing its gradient)

iii. Diffusion Na⁺ current must be inward, so …

iv. must have [Na⁺]_out > [Na⁺]_in

(4) Dendrite

a. Permeability increased by neuro-transmitter-sensitive gates at synapse

b. Concentration gradients drive additional ionic current, reducing membrane charge imbalance

c. Drives V less negative/more positive at synapse;

d. Same for V nearby, since dendrite is conducting

e. V, I, decay quickly (exponentially, length constant ≈ 1 mm)

f. Vs add up at beginning of axon, depolarizing it to a degree dependent on stimulation

(5) Consequences: axon slightly depolarized (-75 mV ≲ V ≲ -50 mV) [indirectly by dendrites]

a. Permeability of membrane to Na+ increases slightly

b. Resting concentration gradient drives a little more Na+ in through membrane

c. Reduces net -charge on inside surface of membrane slightly

d. Drives V slightly less negative/more positive

e. Decreases slightly inward electromotive force on K+ in membrane

f. Reduces electromotive resistance to outward K+ diffusion

g. Increases outward K+ flow slightly, counteracting increased inward Na+ flow:

h. Negative feedback

(6) Consequences when moderately depolarized (when V ≳ -50 mV)

a. Resting concentration gradient drives Na+ in through elevated permeability

b. Reduces net -charge on inside surface of membrane

c. Drives V less negative/more positive

d. Increases permeability of membrane to Na+ still further:

e. positive feedback

f. Eventually inflow of Na+ causes net inflow of total current (driven by Na+ concentrations)

g. Na⁺ permeability rapidly increases: spike

h. V repolarizes because

i. Na⁺ channels spontaneously de-activate after being open awhile

ii. Voltage-sensitive K+ channels open (delayed relative to Na+ channels), K+ current cancels

(7) Propagation

a. As in dendrites, V changes at one point on axon cause V changes at adjacent points

b. large V change nearby :

i. ‘down the axon’: opens Na⁺ gates, re-initiates spike

ii. ‘up the axon’: Na+ gates deactivated, K+ gates still open, so no spike

c. Myelin

i. plugs leaky membrane, reduces capacitance: speeds passive propagation (τ = 1/RC)

ii. gaps — nodes of Ranvier — have gates where spike re-initiates

(8) Gates

a. Complex molecules with multiple states

b. Some states ‘open’ a channel, significantly increasing conductivity g

c. States described by variables ‘n, m, h’ in Hodgkin-Huxley gate model — transitions between states is stochastic: probabilistic model to be discussed later in course.

d. Transition probabilities depend on voltage across membrane (unequal charge density in molecule responds to electric field)

Introduction to Cognitive Science for Mathematical Scientists

Cognitive Science 050.313/613

Neural Currents IV: Equilibrium Analysis

September 5, 2003

(1) Passive current flow: Nernst-Planck Equation

a. Given:

i. Y, an ion of valence z Î ℤ (charge q = ze)

ii. a membrane permeable to Y , with different concentrations [Y]_in , [Y]_out

b. Two pressures driving flow of Y across the membrane:

i. Electric field causes drift flux (in molecules/sec-cm², not charge/sec-cm²)

J_drift = -m z [Y ^z] ÑV — m = mobility

ii. Concentration gradient causes diffusion flux; Fick’s Law:

J_diff = -D Ñ[Y ^z] — D º diffusion coefficient

D = (kT/e) m — Einstein (1905; Brownian motion)

iii. Net result: Nernst-Planck equation

(2) Equilibrium potential of an ion Y^z , V_m: Nernst Equation — no net Y current across membrane

a. V_m º V_in - V_out , the voltage difference where J = 0 (J_diff and J_drift cancel — equilibrium); the resting potential of Y_i:

— Nernst Equation

≐ (62 mV / z) log₁₀ ([Y]_out/[Y]_in) — at body temperature, 37°

Derivation (Assuming constant E = V_m/Dx):

0 = J_diff + J_drift Þ -J_diff = J_drift Þ

D d[Y]/dx = -m z [Y] dV/dx = -m z [Y] E

d[Y]/dx = -c [Y] c º mzE/D = zEe/kT

[Y] = k e ^{–c x}

[Y]_out/[Y]_in = e ^{–c Δx}

ln([Y]_out/[Y]_in) = -c Dx = -zeEDx/kT

(kT/ze) ln([Y]_out/[Y]_in) = -EDx = V_m

b. Since conductance is defined by I_drift = gV and this is cancelled by I_diff at the resting potential E_i:

I_{diff i} º gE_i

This is (approximately) constant because the ion concentrations are (approximately) constant.

c. For two permeable ions Yⁿ and Z^p (because there is one common V_m for all ions):

— Donnan equilibrium

(3) Constant Y current through membrane: Goldman-Hodgkin-Katz (GHK) model

a. Ions flow through cross-membrane protein molecules with aqueous pores

b. Assumptions (for simple pores — not complex, V-sensitive channels)

i. Ion currents across the membrane obey the Nernst-Planck equation

ii. Ions do not interact with each other

iii. E is constant in the membrane

c. GHK current equation:

where P_i º μ_iβ_ikT/le, b_i º water-membrane partition coefficient for Y_i , α º eV/kT, and

Derivation: I_i = z_ie J_i (I_i in coulombs/cm²; J_i in molecules/cm²; l º membrane thickness)

I_i = a_i[Y_i] - b_i d[Y_i]/dx Nernst-Planck: a_i º -m_iz_i²e ∂V/∂x = m_iz_i²eV/l b_i º m_iz_ikT

(I > 0 by def. when I flows out through the membrane)

-b_i d[Y_i]/dx = I_i - a_i[Y_i] = y y º I_i - a_i[Y_i] change variable to y

dy/dx = dI_i/dx - a_i d[Y_i]/dx

= 0 - a_i (-1/b_i) y

= a_i (y/b_i)

a_i/b_i = (z_ie/kT)V/l = z_iα/l

k = y(0) = I - a[Y]_x=0 [Y]_x=0 = b_i[Y]_in [Y]_x=l = b_i[Y]_out (definition of b_i)

I_i - a[Y]_x=l = y(l)

where a_iβ_i = m_iz_i²β_ieV/l = P_iz_i²eα

à Note: Can view as the net result of I_out = γ[Y]_in , I_in = γe^–zα [Y]_out, γ º Pz²eα/(1 - e^–zα)

d. Multiple ions, each I_i = constant, I_total = 0 (equilibrium)

i. I_total = Σ_i I_i , where I_i is given in (c), by the assumption that currents are independent

ii. The resting potential of a cell with ions Y_i º {K⁺, Na⁺, Cl^–} and membrane permeabilities P_i (in cm/sec: [flux of Y_i , in molecules/sec-cm²] = P_i Ñ[concentration of Y_i, in molecules/cm³])

— GHK voltage equation

Derivation: For K⁺, Na⁺, and Cl^–, z² = (±1)² = 1. Want V = αkT/e = (kT/e ) ln (e^α)

0 = I_total = I_K + I_Na + I_Cl

à Note 1: with a single ionic species, I_total = I_i , this becomes the Nernst Equation

à Note 2: one of the ion ‘species’ can be an ion pump Þ effect of pump on V_rest (≈10%)

References:

Kandel, Eric R., Schwartz, James H., and Jessell, Thomas M. 1991. Principles of Neural Science. New York: Elsevier. Third Edition, Chapter 6.

Johnston, Daniel, and Wu, Samuel Miao-Sin. 1995. Foundations of Cellular Neurophysiology. Cambridge, MA: MIT Press.

Introduction to Cognitive Science for Mathematical Scientists

Cognitive Science 050.313/613

Neural Currents V: Propagating Action Potentials

September 5, 2003

(4) Parameters

a. a = radius of cylinder; A = pa² = cross-sectional area of cylinder

b. R_i = specific intracellular resistivity (Ω-cm)

R_i^Dx º resistance to axial flow of a length Dx of cable = R_i Dx/A (Ω = Ω-cm ´ cm/cm²)

c. R_m = specific membrane resistivity (Ω-cm²)

R_m^Dx º resistance to trans-membrane (radial) flow of a length Dx of cable = R_m/2paDx (Ω = Ω-cm² / cm´cm)

d. C_m = specific membrane capacitance (F/cm²)

C_m^Dx º capacitance of membrane of a length Dx of cable = C_m2paDx (F = F/cm² ´ cm²)

e. J_i = interior (axial) current density

I_i = J_i A º interior (axial) current

f. J_m = trans-membrane (radial) current density

I_m^Dx º radial current along a length Dx = J_m 2