Nervous System

Question 1

(a) (3) Nernst equation: $E_{ion} = \dfrac{RT}{zF}\ln\dfrac{[\text{ion}]_{out}}{[\text{ion}]_{in}}$ (1). An equilibrium potential exists because ion movement is driven by two opposing forces — the chemical (concentration) gradient and the electrical gradient (1); at $E_{ion}$ the electrical force exactly balances the diffusion force so net flux = 0 (1).

(b) (4) Use $\dfrac{RT}{F} = \dfrac{8.314\times310}{96485} = 0.02671\ \mathrm{V} = 26.71\ \mathrm{mV}$; $z=+1$.

• $E_K = 26.71\ln(5/140) = 26.71\times(-3.332) = -89.0\ \mathrm{mV}$ (2)
• $E_{Na} = 26.71\ln(145/15) = 26.71\times(2.269) = +60.6\ \mathrm{mV}$ (2)

(c) (6) GHK: $V_m = \dfrac{RT}{F}\ln\dfrac{P_K[K]_o + P_{Na}[Na]_o + P_{Cl}[Cl]_i}{P_K[K]_i + P_{Na}[Na]_i + P_{Cl}[Cl]_o}$ (note Cl inverted because $z=-1$) (2). Numerator: $1(5) + 0.04(145) + 0.45(10) = 5 + 5.8 + 4.5 = 15.3$ (1). Denominator: $1(140) + 0.04(15) + 0.45(110) = 140 + 0.6 + 49.5 = 190.1$ (1). $V_m = 26.71\ln(15.3/190.1) = 26.71\times(-2.521) = -67.3\ \mathrm{mV}$ (2).

(d) (4) Passive: the membrane is far more permeable to $\mathrm{K^+}$ than $\mathrm{Na^+}$ at rest (leak K channels), so $V_m$ sits near but not at $E_K$; the small $\mathrm{Na^+}$ leak pulls it positive of $E_K$ (2). Active: the $\mathrm{Na^+/K^+}$-ATPase pumps 3 Na out / 2 K in per ATP, restoring gradients and contributing a small electrogenic hyperpolarising current, maintaining the resting state long-term (2).

(e) (5) $E_K$ depends on the log of the ratio; doubling $[K]_o$ changes ratio 5/140 → 10/140 (2). $E_K^{new} = 26.71\ln(10/140) = 26.71\times(-2.639) = -70.5\ \mathrm{mV}$ (2). $E_K$ becomes less negative (depolarised by ~18.5 mV), moving resting potential toward threshold → increased excitability; in hyperkalemia this causes cardiac arrhythmias / eventual inactivation of Na channels (1).

Question 2

(a) (6) — 1.5 marks per phase (½ description, ½ channel + ½ shape):

• Resting $-70$ mV: voltage-gated Na/K closed, leak channels set potential.
• Depolarisation: voltage-gated $\mathrm{Na^+}$ channels open, Na influx, rapid rise to ~+30 mV.
• Repolarisation: Na channels inactivate; voltage-gated $\mathrm{K^+}$ channels open, K efflux, potential falls.
• Hyperpolarisation (undershoot): slow K channel closure overshoots below rest, then leak restores. Labelled sketch with mV/ms axes and threshold line (~$-55$ mV).

(b) (4) Absolute refractory period: Na channels are inactivated (ball-and-chain), cannot reopen regardless of stimulus (2). Relative refractory period: K channels still open + some Na recovered → larger-than-normal stimulus needed (1). Because the region just fired is refractory, the AP can only propagate forward into un-refractory membrane → unidirectional (1).

(c)(i) (3) Unmyelinated: $t = 1.0/25 = 0.0400\ \mathrm{s} = 40.0\ \mathrm{ms}$ (1.5). Myelinated: $t = 1.0/120 = 0.00833\ \mathrm{s} = 8.33\ \mathrm{ms}$ (1.5).

(c)(ii) (4) $v \propto \sqrt d \Rightarrow v_2/v_1 = \sqrt{d_2/d_1}$. $120/25 = \sqrt{d_2/500}$ (1) → $(4.8)^2 = d_2/500$ (1) → $d_2 = 23.04\times500 = 11520\ \mu\mathrm{m} \approx 11.5\ \mathrm{mm}$ (1). This is biologically infeasible — an 11.5 mm axon is enormous; myelination achieves high velocity with a 20 µm axon, a ~500× space saving (1).

(d) (5) — award for: increment (1), threshold test (1), spike+reset (1), refractory counter (1), loop structure (1).

def integrate_and_fire(inputs, V_thresh=-55, V_reset=-70, V_rest=-70, refrac=2):
    V = V_rest
    spikes = []
    ref_timer = 0
    for t, I in enumerate(inputs):
        if ref_timer > 0:          # refractory: ignore input
            ref_timer -= 1
            V = V_reset
            continue
        V += I                     # integrate synaptic input
        if V >= V_thresh:          # fire
            spikes.append(t)
            V = V_reset            # reset
            ref_timer = refrac     # enter refractory period
    return spikes

Question 3

(a) (5) — 1 mark each: AP reaches terminal → opens voltage-gated $\mathrm{Ca^{2+}}$ channels; $\mathrm{Ca^{2+}}$ influx triggers vesicle fusion with presynaptic membrane; neurotransmitter released by exocytosis into cleft; diffuses and binds receptors on postsynaptic membrane; opens ion channels → EPSP/IPSP; then neurotransmitter removed (reuptake/enzyme degradation).

(b) (5) Components (label any 5): receptor (muscle spindle) → sensory (afferent) neuron → dorsal root → spinal cord (integration centre) → motor (efferent) neuron → effector (muscle) (3). No brain required because integration occurs at the spinal cord synapse — faster, protective (1). Monosynaptic stretch reflex has sensory + motor neurons only (no interneuron); withdrawal reflexes add interneurons (1).

(c) (6) — 1 mark per correct cell:

Target-organ neurotransmitter: sympathetic → noradrenaline (norepinephrine) at most targets; parasympathetic → acetylcholine (2 marks for the two transmitters).

[
  {"claim":"E_K approx -89.0 mV", "code":"RT_F=8.314*310/96485*1000; E_K=RT_F*log(Rational(5,140)); result = abs(float(E_K)-(-89.0))<0.5"},
  {"claim":"E_Na approx +60.6 mV", "code":"RT_F=8.314*310/96485*1000; E_Na=RT_F*log(Rational(145,15)); result = abs(float(E_Na)-60.6)<0.5"},
  {"claim":"GHK V_m approx -67.3 mV", "code":"RT_F=8.314*310/96485*1000; num=1*5+0.04*145+0.45*10; den=1*140+0.04*15+0.45*110; Vm=RT_F*log(num/den); result = abs(float(Vm)-(-67.3))<0.6"},
  {"claim":"New E_K with 10mM outside approx -70.5 mV", "code":"RT_F=8.314*310/96485*1000; E=RT_F*log(Rational(10,140)); result = abs(float(E)-(-70.5))<0.6"},
  {"claim":"Diameter for 120 m/s unmyelinated approx 11520 um", "code":"d=(120/25)**2*500; result = abs(d-11520)<50"}
]