Lecture 2, Cellular Communication: AP Propagation & Nervous System

• Ref: Pf. Kang's SNUCM Lectures

1. Nervous System

2. Cable Equation : Action Potential Propagation

• KCL, KVL: Kirchhoff's Current / Voltage Law
  

가정
1. 1D model
2. Inside: 단위길이당 저항 : $R_c$
3. Outside: 저항 : $0$
4. 세포막 단위길이 당 Capacitance : $C_m$
5. 세포막 단위길이 당 Cunductance : $\frac{1}{R_m}$ (ions leaky chn.)

2-1. KCL

• dx: 1D 세포막 길이

  • 축전용량: dx에 비례 (bc. 세포막 넓이에 비례)
  • 저항: dx에 반비례 (bc. 세포막 넓이에 반비례)
    $C = C_m \, dx, \quad R = \frac{R_m}{dx}$

• KCL at #1

  $i + i_c + i_{R_m} = i + di$

  Thus,

  $di = i_c + i_{R_m}$

  • 축전기

    $Q = C \, V = C_m \, dx \, V$
    $i_c = \frac{dQ}{dt} = \frac{d}{dt} \left( C_m \, dx \, V \right) = C_m \, dx \, \frac{dV}{dt}$

  • 저항

    $i_{R_m} = \frac{V}{R_m} \, dx$

  • => KCL 대입

    $di = C_m \, dx \, \frac{dV}{dt} + \frac{V}{R_m} \, dx$

최종 식 (1)

$R_m \frac{d i}{dx} = R_m \, C_m \, \frac{dV}{dt} + V$

2-2. dV eq.

• dV

  $dV = i \, R_c \, dx$

  Thus,

  $i = \frac{1}{R_c} \frac{dV}{dx}$

• 최종 식 (1)에 대입

  최종 식 (2)

  $R_m \frac{1}{R_c} \frac{d^2 V}{dx^2} = R_m C_m \frac{dV}{dt} + V$

• 상수 정의

  • 길이 상수 (cable space constant) ~ 150um
    $\lambda_m^2 = \frac{R_m}{R_c}$
    즉,
    $\lambda_m = \sqrt{\frac{R_m}{R_c}}$
  • 시간 상수 (membrane time constant) ~ 8.4ms
    $\tau_m = R_m C_m$

2-3. Cable equation

Cable equation

$\lambda_m^2 \frac{\partial^2 V}{\partial x^2} = \tau_m \frac{\partial V}{\partial t} + V$

2-4. AP Propagation

• Steady State: $dV/dt = 0$

  $\lambda_m^2 \frac{d^2 V}{dx^2} = V$

  즉,

  Steady State:

  $V(x) = V_0 e^{-\frac{x}{\lambda_m}}$

• 가정

  • $V_0 = 100mV$
  • AP threshold $= -60 + 20 (mV)$
  • $\lambda_m=150\mu m$

• Maximum distance (AP peak-to-peak)

  • 계산 방법
    $V(x) \geq 20$
    즉,
    $100 e^{-\frac{x}{150}} \geq 20$
    $x \leq 150 \ln{5} = 240(\mu m)$
  • Maximum AP p2p distance는$\lambda_m$에 비례

• Nerve의 diameter가 커질수록, $R_c$ 감소 -> $\lambda_m$ 증가 -> AP propagation 속도 증가(p2p 거리 증가)

AP 전파 속도
1. $\lambda_m$에 비례
2. Nerve Diameter에 비례

2-5. Ranvier nodes

• Larger $R_m$

2. Cellular Communication

2-1. Synapses

• Chemical Synapse
  

• Electrical Synapse

  • ex. Cardiac muscle cells
    

2-2. Neuromuscular junction

• Neuromuscular junc.
  • {1 motor neuron} for {1 muscle cell} synapse
  • $\sum$EPSP = AP (in post synaptic neuron)
    - Temporal Summation
    - Spatial Summation
    

2-3. EPSP & IPSP

• Excitatory postsynaptic potential (EPSP)
  • Post synaptic cell: Depol
• Inhibitory postsynaptic potential (IPSP)
  • Post synaptic cell: Hyperpol
    

3. Cardiac Muscle cells: Heart

3-1. AP in Cardiac Muscle Cells

• Ca2+ 차이

3-2. AP in SAN (SA Node) Cells

Nervous system ~ electrical circuit (bioelectronics system)