Test 1: Computational Neuroscience

Why would we focus on the soma when trying to understand a neurons computation?

where inputs are combined into one decision variable and where spike threshold

lives, so it captures the core input→decision→spike computation.

What is a neuron's ‘computation’, and how is it represented

electrically?

The transformation of input current patterns into an output spike train / firing rate.

Represented by V(t) dynamics: C_m dV/dt = −ΣI_ion + I_ext.

State the 3 main ions for resting potential and their relative

concentrations.

1. Na+ — higher outside, lower inside

2. K+ — higher inside, lower outside

3. Cl- — higher outside, lower inside

Why does ion movement create current, and why can't ions cross

freely?

Current = net charge flow per time; moving charged ions is a current.

The lipid bilayer core is hydrophobic; ions are charged/hydrated 

What is the Nernst equilibrium

The voltage at which electrical and chemical driving forces on that ion balance → zero net flux.

Define membrane potential V_m and how it differs from a single ion's E_ion.

1. V_m = the actual net voltage from ALL open channels combined.

2. E_ion = the voltage where just ONE ion's current alone is zero.

Why does V_m settle near −70mV even though no single E_ion equals −70mV?

V_m settles where the SUM of all ionic currents = 0, not where one ion's current alone is zero.

What determines the magnitude and direction of I_ion =

g_ion(V_m−E_ion)?

1. Magnitude: g_ion (openness) × driving force (V_m−E_ion) (distance from equilibrium).

   
   

2. Direction: sign of driving force alone — V_m > E_ion → outward (+); V_m < E_ion → inward

   (−).

Inward current depolarises V_m — how do driving forces/currents change?

1. V_m−E_K becomes more positive as V rises → outward K+ current increases.

2. V_m−E_Na becomes less negative → inward Na+ driving force shrinks (at fixed

conductance).

Why is there a time lag between applying current and V_m settling?

The membrane is an RC circuit — the capacitor can't change voltage instantly; charge must accumulate via finite current while some current also leaks away

What does it mean for an ion channel to be 'voltage gated'?

Its open probability (conductance) depends on V_m rather than being fixed.

Which two ions have voltage-gated channels in the standard HH model?

Sodium (Na+) and Potassium (K+).

Relationship between an ion channel's state and its conductance.

Effective conductance = max conductance (all channels open) × fraction currently open

What happens to voltage-gated channels when V_m is near rest?

1. m (Na+ activation) ≈ 0

2. n (K+ activation) low.

3. Both g_Na and g_K small → leak conductance dominates

What does it mean for a membrane to 'depolarise'?

V_m becomes less negative (moves toward 0/positive)

What happens to voltage-gated channels as membrane depolarises?

1. m rises rapidly (tiny τ_m)

2. g_Na spikes up

3. large inward Na+ current (driving force still large, E_Na≈+50mV)

4. further depolarises V_m: positive feedback = upstroke.

Why does HH need three gating variables (m, n, h) instead of one?

Two channel types (Na+, K+) with independent kinetics; Na+ current is transient (rises then falls even if V stays high) — impossible with one monotonic gate.

Which HH variables are fast (→V) and which are slow (→W)?

1. Fast → V: membrane potential V, plus m (tiny τ_m, tracks V almost instantly).

2. Slow → W: h (Na+ inactivation) and n (K+ activation), both much larger time constants.

Why does current I appear only in the fast equation dV/dt =V−V3/3−W+I?

 Injected current directly charges the membrane capacitance

Does a limit cycle in the phase diagram mean spiking or resting?

Spiking. A limit cycle is a closed trajectory the system cycles around forever instead of settling to a fixed point

Why doesn't a neuron fire when I is too low, or too high?

1. Too low: fixed point stays on the left, stable branch — perturbations decay back to rest, no threshold crossing.

2. Too high: fixed point moves onto the right, stable branch— depolarisation block, no cycling.

Define a bifurcation

A qualitative change in system behaviour as a parameter is varied through a critical value

HH→FN: what's lost ?

Lost: explicit ion-channel identity/interpretability — no separate Na+ activation/inactivation, no way to simulate blocking a specific channel

HH→FN: why do we want the simpler system anyway

 2-D systems can be fully analysed via phase-plane/nullclines:

Cheaper to simulate, yet still captures threshold qualitatively.

Map leak / integrate / fire / reset onto Algorithm 1 (LIF)

1.  −(V−V_rest): Leak

2. R_m·I(t) : Integrate

3. V≥V_th then S(t)←1: Fire

4. V←V_reset:Reset

Define the rheobase of a neuron.

The minimum constant input current needed to produce firing/spike

What do x and ReLU(x) represent?

What is the difference between LIF and ReLU?

LIF: Gives spike and firing rate

ReLU: Gives continuous activation

Why are ANNs often interpreted as rate-coding models?

Each ReLU ANN neuron outputs one continuous scalar per pass (not a spike train); information is carried by the value's magnitude 

what does HH capture?

explicit biophysics:

1. real channel types

2. voltage/time-dependent conductances

What does FN capture?

lumps channels into fast V / slow W:

1. keeps fast/slow structure and emergent spiking

2. loses channel-specific detail

What does LIF capture?

one passive linear equation + an imposed threshold-reset rule:

1. no channel mechanism

2. no fast/slow split

3. spike is a discrete event not an emergent voltage excursion

Which part of the neuron carries the AP from soma to synaptic terminal?

The axon.

Define a neurotransmitter.

A chemical messenger released from presynaptic vesicles that diffuses across the synaptic cleft and binds postsynaptic receptors

What does it mean for a synaptic input to be excitatory or inhibitory?

1. Excitatory: depolarises toward threshold—E_syn above resting/threshold

2. Inhibitory: hyperpolarises or shunts — E_syn at/below resting potential (e.g. GABA/Cl-)

How can changes in receptor number alter g_syn(t)?

1. More receptors inserted→ more current flows for the same neurotransmitter release → higher effective max synaptic conductance.

2. Fewer/lower-efficacy receptors → lower g_syn(t). Because this ceiling itself can change over time

Explain 'fire together, wire together' and how STDP refines it

Basic Hebbian idea: persistent pre/post correlation → synapse strengthens.

STDP adds precise causal timing: pre-before-pos

How does coincident pre/post activity cause a persistent strength change?

Coincident release + strong postsynaptic depolarisation→large Ca2+ influx → signalling cascades (e.g. CaMKII) → more AMPA receptors inserted/higher efficacy→ persistently raised g_syn (LTP)

Which biological synapse property is most closely abstracted by ANN weights?

The (maximum) synaptic conductance / connection strength

Primary problem with pure Hebbian learning, and how BCM fixes it

Pure Hebb: only ever strengthens (positive feedback) → unbounded weight growth, destabilises network.

BCM: sign of plasticity depends on y vs sliding threshold θ_m — y<θ_m → LTD, y>θ_m →LTP — bidirectional and self-limiting.

Why is BCM considered a homeostatic learning mechanism?

θ_m slides with the neuron's own recent average activity avg(y): high avg(y) → θ_m rises →harder to potentiate, easier to depress — pulls activity back down; low avg(y) → opposite

Effect of ↑ extracellular ion concentration on E_ion?

Increases (more positive). Larger [ion]_out/[ion]_in ratio → larger ln term → larger E_ion for a cation.

Which term is the capacitive current in the RC membrane equation?

C_m·dV_m/dt

Role of W in FitzHugh-Nagumo relative to fast variable V?

It acts as a slow recovery variable representing K+ activation and Na+ inactivation

Major limitation of standard unconstrained linear Hebbian

learning?

Weights can grow unbounded toward infinity due to positive feedback.

How does BCM prevent runaway synaptic weight growth?

By introducing a dynamic threshold θ_m that scales with average postsynaptic

activity.

T/F θ_m shifts upward when avg(y) is high, making further LTP harder?

True. The homeostatic sliding-threshold mechanism.