ID vs 2D

Cellular biophysics and modeling

Main topic

Membrane potential dynamics

Action potentials (excitability)

Action potentials (excitability)

Hodkin huxley discovered there 2 currnets in action potentials

peaks at Na becomes given a voltage and also where I will eventually steady given voltage

peaks at Na becomes given a voltage and also where I will eventually steady given voltage

Na+, early inward current

Na+, early inward current

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changing Na concent only saw change in the early inward curent

the 2 blips at +70 and +90 mv are evoke at volatges above nerst potential

K+ , delayed outward current

separation and subtraction of ionic currents

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LAST THREE MINUTES OF 9.1????????????embracing currents can be isolated via ion subsists ion and digital subtraction

slope g / E is equilib / current is dep on driving force (E--> eq disparity / once cross x, start leaving cell vs entering /

slope g / E is equilib / current is dep on driving force (E--> eq disparity / once cross x, start leaving cell vs entering / there's a curve when whntering is highly pref

perisistent

perisistent

actiavtion by depolarization

actiavtion by depolarization

I Kv

I Kv

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once reaches m inf all conduxctance is available (all channels open)

I CaV depolarization activated Ca2+ current

activation by hyperpolarization

activation by hyperpolarization

inward recitfying potassium current Kir

inward recitfying potassium current Kir

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always open until threshold or

physiological range

physiological range

Ih or Isag

Ih or Isag

delayed-reticifier

delayed-reticifier

I potassium - DR = conductance m^n (Driving force)

I potassium - DR = conductance m^n (Driving force)

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opposite shape for hyperpolarization

membrane bistability

3 times interescts x axis, current is 0

currents on either eside of the 2 steady states are resorative (positive conductance) an perturbation would restore to Ek. membrane would fall back down unless you reach threshold

stable at 2 different Vs and a transient current can switch you between the two

transient (happens when inactivate)

transient (happens when inactivate)

I T

I T

shifting Iapp can chahnge stabilities

hyteresis

state stapcs vs parameter (Iapp)= bifurfaction diagram

3 points at which intersect

equal and opposite so cancel out

restoraive restorative

regernative regernative

mathematically stable unrealistic because system cant maintain like a pen standing upright

restorative e

capacitive transient

I ion - Activation of K+ current when voltage back -10 --> -70, low driving force because ion channels decativate

I ion - Activation of K+ current
when voltage back -10 --> -70, low driving force because ion channels decativate

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only flows at instant voltage changes

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conductance is nonnegative NOT CURRENTn gates 4 gates so the moventnt of any one 4 gates has ability to move close gate since 4 particles must be in a particular configuration

m∞

Na+ activation

Na+ activation

deactivation- falls back to 0 state bc evoked stimulus is removed

deactivation- falls back to 0 state bc evoked stimulus is removed

voltage dependant rate constant

α: closed --> open

β : open --> closed

gating variable (m)

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(bc channels have voltage dependant gates)which also is why they inactivate after same evokedvoltage too

(1-m) ⇌ m

m[0,1]

How close is M to its steady state value? How fast is scaled by time constast (big-slow vv)

How close is M to its steady state value?
How fast is scaled by time constast (big-slow vv)

M ss

M ss

Tau

Tau

dm/dt = α(1-m) - βm

= α-m(α+β)

dx/dt = a + bx

dc/dt = j - kc

Cdv/dt = I app - gk ( V - Ek)

antiderivative

antiderivative

xss = a/b

τ = 1/ b

order of magitigude
(10x) faster, react to changes

h∞

I na inactivation

I na inactivation

n∞

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capaticence = ability to store and separate charge = lipid bilayer because they are impermeatble

extensive vs intensive

intensive

memebrane reistivity Rm = 10000 Ωc^2

capacitance

extenstive

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x^2 when x*2 factor by 4

membrane resistance Rm = Rm/ SA

4πr^2

membrane capacitance

ion Channels

transmenbrane potential

GHK current density equation

GHK current density equation

zV/Vθ

vθ = RT/F

for Na+

for Na+

reversal potential = nernst potential where current reverses

reversal potential = nernst potential
where current reverses

ratio defines amount of retification

ratio defines amount of retification

opposite for K+ and flip out and in anions

opposite for K+ and flip out and in anions

inwardly rectifying K+ potassium current

inwardly rectifying K+ potassium current

zF

Ps

[S]i

[S]out

predicts non lineariity bc mutiple variables inviolved

ions move because diffsion and drift (C and electromotove)

Vm = φin - φout

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as +ions that are higher [outside] are allowed in move in, will do so nonlinearly because gradually will want to move in less because electromotive force that is opposing the still existing concentration graduient

nernst EQUILIBRIUM potential

nernst EQUILIBRIUM potential

Goldman hodkin katz (ghk) voltage equatoion

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note that cl has flip i/o so that we don't need to use z +1 * +1 = -1 ^ 2 1*2 = -1^2

assume Pcl<<< PK (we did same for Ca++)

α = PNa/PK

a<<<1 and so [Na]i<<<[K]i therfore

a<<<1 and so [Na]i<<<[K]i therfore

constant field theory

current voltage relations

current voltage relations

I cap

I cap

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for every one charge that enters, one leaves from other side through nothing actually really passes

Q = CVm

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C = capacitance

Icap = dQ/dt = C (dVm/dt)

Ires

Ires

Vres

IresR

R=1/gk

Ires/ gk

φ* - φout

Ek = φin -φ*

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special case of ghk current equationGK -- current ---> voltage ---> nernst

Vm = Ek + Vres =φin - φ* + φ* - φout

Vm = Ek+ Ires / gk

Ik = gk(Vm-Ek) chord conductance equation form

Ik = gk(Vm-Ek) chord conductance equation form

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current over resistor (ion channel)

Current (C/t) = A

Current (C/t) = A

- / INWARD

+ / OUTWARD

hyperpolarizing

I cap

- ion out --> in

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ex GABA cl- outward currnetsCl- entering

+ ions in --> out

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Na + or Ca2+ leaving

depolarizing

applied current

kirchoffs current law =I res + I cap - I app = 0

Ires

I cap

Iapp

current balance equation

C(dV/dt) = Iapp - Imem

Imem = Σion

Ileak
Passive cell

Iion=
gk(Vm-Ek)

exponential relaxtation

dv/dt= [(Iapp +gkEk)/C] - (gk/C) V phase diagram

dv/dt= [(Iapp +gkEk)/C] - (gk/C) V phase diagram

Ileak +Ica
(Bistable)

Ileak + Ina + Ikv
Hogdkin-Huxley

Experimental Recording Methods

current clamp recording

voltage clamp recoding FIXED VOLTAGE, MEASURE CURRENT

voltage clamp recoding
FIXED VOLTAGE, MEASURE CURRENT

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changing salt concentrations

axons dont have other than Na + an k+ channels

1 - Measure

2 - Compare

3- Correct

4 - Inject

5 - Monitor

applied current adjusted so that V = Vcommand

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Depolarizing or depolarizing voltageChange I app very fastto create steps (pulses)instantanous

(Depolarizaing) Pulses

(Depolarizaing) Pulses

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steps to higher voltages)

dv/dt = 0
0 = Iapp - I mem
(C +) I mem = Iapp

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What happend to C ?????

Main topic

Main topic

Main topic

Main topic

ODE Modeling

phase diagrams

^ f(x) on y and state variable x on x axis
(derivative as a function of underived function)

idenitfy critical points, cross at x axis is css

determine stability by lookin at a fllow directions

preturb off its xss: n = x -x*

dn/dt= d/dt (x-x*)= f(x) - f(x*)

dn/dt = f(x) - 0

= f(x) = f(x* + n)

via expansion taylor
dn/dy = f(x*) + nf'(x*) + O(n²)

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very small perturbation so third term is negligible

= 0 + nf' (x*) + 0

derivative of f(x*) is not 0 bc this is f''

> 0 unstable

speeding away

< 0 stable

speeding toward

d/dx {f(xss)} = 0 (y axis)

phase plane analysis

bifurcation

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Floating topic

restorative (+slope) regenerative (- slope)

restorative (+slope) regenerative (- slope)

regenerative

negative conduction

restirative

hyperpolarization (+ -V) ion (+ +V)

inward / outward

dv/dt = Iapp (t) - Iion (V)
Iapp (t) = C dv/dt + Iion (V)

Imem
= Cdv/dt + Iion (V)

Imem= Icap + Iion