Quantitative Neuroscience - Final Exam (Updated from exam 1)

MATH 4350: Quantitative Neuroscience at OSU w/ Janet Best

The membrane potential is determined by…

membrane conductances (g_i), gating variables, and corresponding reversal potentials (E_i)

Neurons are ____ because the currents depend on…

excitable; the membrane potential and time.

The most accepted description of kinetics of voltage-sensitive conductances is…

the Hodgkin Huxley gate model

Na⁺ and Ca²⁺ concentrations are generally higher ____, so voltage-gated activation of these channels results in ____ currents that ____ the membrane potential.

outside the neuron; inward; depolarize/increase

K⁺ concentrations are generally higher ____, so voltage-gated activation of these channels results in ____ currents that ____ the membrane potential.

inside the neuron; outward; hyperpolarizes/decreases

Gating variables m and n are ____.

activating (monotonically increasing with V)

Gating variable h is ____.

inactivating (monotonically decreasing with V)

After the Na channel has inactivated, it has to ____ before the neuron can fire another action potential.

deinactivate

The Hodgkin-Huxley equations are a ____ model.

4D (conductance-based ODE)

The Morris-Lecar equations are a ____ model.

2D

We use ____ and ____ to understand observed dynamics.

plane analysis; bifurcation diagrams

Tonic spiking corresponds to a ____ limit cycle.

stable

Class 1 excitability

SNIC bifurcation; the neuron can fire with arbitrarily low frequency (continuous f/I curve)

Class 2 excitability

Hopf bifurcation; the neuron can only fire within a certain frequency range (discontinuous f/I curve)

For the ML equations with homoclinic parameter set, the onset of spiking is due to a ____ bifurcation.

The neuron jumps to spiking at a ____ (discontinuous f/I curve).

saddle-node

positive frequency

In the (∆,τ) plane, the left side corresponds to…

saddle/unstable fixed points (two real eigenvalues of opposite sign)

In the (∆,τ) plane, the quadrant for stable fixed points is the ____ quadrant.

bottom, right

Firing becomes arbitrarily ____ as you approach a SNIC or homoclinic bifurcation

slow (frequency approaches zero)

Signs of ∆ and τ for a stable fixed point

∆ > 0

τ < 0

A Hopf bifurcation occurs when…

there are a pair of pure imaginary eigenvalues

Criteria for node fixed points

real eigenvalues of the same sign

Criteria for saddle fixed point

real eigenvalues of opposite sign

A saddle-node bifurcation occurs when…

a saddle fixed point and a node fixed point merge

SNIC bifurcations occur when…

a saddle-node occurs on an invariant circle

Bursting requires at least ____ time scales.

3

The A-current (outward, inactivates) converts a cell from ____ to ____ excitability.

class 2, class 1

The M-current (outwards, does not inactivate) converts a cell from ____ to ____ excitability (at around g_m = 2).

class 1, class 2

Takens-Bogdanov (TB) bifurcations happen when…

a Hopf bifurcation and a SN bifurcation meet

For changes in ion concentration, flux is ____ for concentrations inside of the cell.

negative

For changes in ion concentration, flux is ____ for concentrations outside of the cell.

positive

Amplifying gating variables

Amplify changes in V through positive feedback

• Activation of an inward current

• Inactivation of an outward current

Resonant gating variables

Resist changes in V through negative feedback

• Inactivation of an inward current

• Activation of an outward current

Cells with fixed points on the left branch are ____.

excitable

Cells with fixed points on the middle branch are ____.

oscillatory

Duty cycle

time spent in active phase/total period

Excitatory synapses have V_Syn ____ V_RMP

greater than (>)

Inhibitory synapses have V_Syn ____ V_RMP

less than (<)

A cell in its silent phase will have a trajectory ____.

on the left branch of the V-nullcline

A cell in its active phase will have a trajectory ____.

on the right branch of the V-nullcline

With inhibitory synapses, C_A is located ____ C₀.

below

With excitatory synapses, C_A is located ____ C₀.

above

In a phase plane for 2 mutually coupled, identical cells, C₀ represents ____.

the V-nullcline for the cell that is not receiving input

In a phase plane for 2 mutually coupled, identical cells, C_A represents ____.

the V-nullcline for the cell that is receiving input

T/F: Both inhibitory and excitatory synapses are able to produce synchronous, almost synchronous, and antiphase firing patterns.

True

An inward current that does not inactivate would show…

increasing rapid spiking.

An outward current that inactivates would show…

delayed spiking (by a steady/not increasing amount).

An outward current that does not inactivate would show…

decreasing (slowing down) spiking.