Hodgkin-Huxley and Neurodynamics

Question 1

What do we mean by stating that the cell membrane is semi permeable?

Answer 1

It means that not everything can pass or diffuse across the lipid membrane.

For example proteins are large molecules that cannot cross the membrane via ion channels. Negatively charged proteins that contribute to the negativity of the resting potential.

Question 2

In a passive cellular membrane that is taken away from V_rest, what are the 2 parameters that contribute to how fast V returns to V_rest?

Answer 2

Membrane capacitance (C_m)

& Leak Resistance (R_leak)

_{Their product is tau, the ‘membrane time constant’.}

dV/dt = (V-Vrest)/(Cm*R)

See project “The Passive Membrane”

Question 3

What are leak channels?

Answer 3

Leak channels are passive ion channels. They are either always open (or they are not sensitive to changes of voltage).

The most frequent leak channels are potassium leak channels, but there are also Chloride and Sodium leak channels.

Question 4

What is conductance? And how is it related to resistance?

Answer 4

Conductance (g) expresses how easy it is for the ions to flow through the cell membrane.

It is the reciprocal of resistance. That is,

g = 1/R

Question 5

How can we model the passive membrane?

Thikn about the process of V tending to V_rest ?

Answer 5

With an ordinary differential equation

For the passive membrane;

dv/dt = -g_leak(V-V_rest)/C

Question 6

What is the driving force when a cell is at equilibrium?

Answer 6

0

( V = Vrest )

Question 7

What do gating variables (m,n,h) represent in HH?

Answer 7

The proportion of ion channels in a certain state (e.g., open/close, active/inactive).

So they take values between [0,1] .

Question 8

The conductance of an ion channel is a function of …

Answer 8

Its maximum conductance and the state of the respective gating variables.

For example for Na+ we have;

Question 9

Why does capacitance (C_m ) affect the rate of change of membrane potential (V)?

Answer 9

Capacitance measures how well the membrane can store charge.

The better the cell can store charge, the slower the charge is released.

A thicker membrane stores less charge for a given voltage difference across it, resulting in lower capacitance.

Question 10

What is a dynamical system? What is a state? What is a state variable? What is a phase space?

Answer 10

A dynamical system - a system whose state evolves over a state/phase space with time according to some fixed rule. What a system will do in the future is a function of where it is now.

A phase space (same as state space) is a set of all possible states (configurations) of a dynamical system.

A state of a system corresponds to a point in the state space.

A state variable is one of the variables that describe the current state of a dynamical system. State variables together fully define the state of the system.

The HH model is a dynamical system with 4 state variables.

Question 11

Regarding the differential equation plotted in this graph. From the left to the right, what are the types of equilibria we observe?

Answer 11

attractor, repellor, attractor

or

stable, unstable, stable

Question 12

Regarding the differential equation plotted in this graph, there are three intersections. What do we call them?

Answer 12

Fixed points.

Question 13

Regarding the following differential equation:

What happens with the equilibria if the F(V) is multiplied by 2?

What happens to the equilibria if we multiply F(V) by -1?

Answer 13

If F(V) is multiplied by 2, the equilibria will not change their stability. However, the particle will move faster through the phase space.

On the other hand, if F(V) is multiplied by -1, the stability of the equilibria will change. We will end up with two repellers and one attractor.

Question 14

In neurodynamics, what is the difference between a parameter and a state-variable?

Give examples of parameters and state variables of a Hodgkin-Huxley neuron model.

Answer 14

The state variable represents the current state of the model.

A parameter is constant over time (or changes slowly).

State variables of HH are: membrane voltage and gating variables of sodium (m, h), and potassium (n)

Some parameters of the HH model: Nernst potentials (E_Na, E_leak, E_K); capacitance, leak conductance, current input, maximum conductances.

Question 15

From the neurodynamics perspective, what causes a spike?

Answer 15

A spike appears due to the imbalance of two opposing dynamics.

1. Positive feedback through sodium conductance (m) which increases the membrane potential
2. Negative feedback through the inactivation of sodium channels and activation of potassium which reduces the membrane potential

Question 16

Consider a neuron model whose membrane potential is governed by the differential equation in the graph below.

Describe what happens with the membrane potential when we perform the following sequence of manipulations:

1. Clamp the voltage to the initial state V(0) and then release it.
2. Add a current step of I = 5pA to the neuron.
3. Next, we remove the added current.
   * What is the name of the phenomenon described above?*

Answer 16

1. Clamp the voltage to the initial state V(0) and then release it. → The rate of change for V(0) is negative, therefore the membrane potential would decrease and stop at the nearest attractor (equilibrium).
2. We add a current step of I = 5pA to the neuron. → When we add a current of 5pA, two equilibrium points are destroyed. We would only be left with one stable attractor. Therefore, if we initialized at V(0), the membrane potential would be attracted toward a higher membrane potential with variable speed.
3. We remove the added current. → If we remove the current step, the system regains two equilibria and we are left with two attractors and one repeller.

This type of change is a bifurcation. A bifurcation is a change such that when a parameter changes, the qualitative structure of the vector field changes dramatically (a fixed point is created or destroyed or changes its stability).

Question 17

How is a model like a neuron? How is it not like a neuron?

Answer 17

A model is a representation of a neuron. It can reproduce spikes that explain and predict action potentials measured in reality. But at the same time, it is not like a neuron, because it is a mathematical abstraction.

Question 18

In Neurodynamics:

A resting state of a neuron corresponds to…

A tonic spiking state of the neuron corresponds to…

Answer 18

…a stable equilibrium…

…a limit cycle attractor

Question 19

What is a trajectory?

What is a flow?

Answer 19

A trajectory is a sequence of states a system can occupy.

A flow is a set of possible trajectories of a system.

Question 20

What is a transient?

Answer 20

A trajectory before the system reaches a steady-state.

Question 21

When can we simplify a multidimensional model?

Answer 21

When (1) introducing details does not contribute to the goal of the model and (2) computational costs outweigh the benefits

Question 22

Name strategies to simplify a neuronal model

Answer 22

0. Reduce number of state variables
1. Remove Ion channels (gates)
2. Simplify Morphology (reduce number of compartments)

Question 23

What are some of the simplification strategies for a neuronal model?

Answer 23

1. Removing an activation variable - when its contribution is too little, when it is too slow, or when you simply want to explore the “what if” situation.
2. Making the activation instantaneous - when time constant tau is very fast.
3. Merging activation gates - when time constants are similar combine the two. For example, the tau of n and h are similar and can be combined into one function.

Question 24

Explain what kind of spiking behavior you see in the image:

Answer 24

Tonic firing - sustained response to a stimulus.

Also, we observe ‘firing rate adaptation’, as after a burst the neuron fires at a regular and lower rate.

Question 25

Explain what kind of spiking behavior you see in the image:

Answer 25

Phasic firing - transient response to a stimulus with spikes at the onset of the stimulus.

Question 26

Classify the following neuron as Type I or Type II and explain.

Answer 26

Type I Neuron: The Integrator

Frequency increases smoothly with increasing current.

Action potentials can be generated with arbitrarily low frequency (firing rate), depending on the strength of the applied current.

Question 27

Classify the following neuron as Type I or Type II and explain.

Answer 27

Type 2 neuron - The resonator

Firing frequency is relatively insensitive to changes in the strength of the applied current.

Question 28

What is the relevance of studying parameter spaces when studying spiking behavior?

Answer 28

By studying the parameter spaces, we understand the dependencies between spiking patterns and ion channel combinations.

The firing patterns of neurons are a consequence of the combinations of their parameters.

Question 29

How should one select state variables to include in a model?

Answer 29

One must attempt to select state variables fully characterizing the natural phenomena under study.

For example, Hodgkin and Huxley selected V, because they could measure it, and m, n, and h variables because to produce an action potential ‘something’ had to be voltage-gated. And via pharmacological manipulations, they intuited that the action potential had to do with Sodium and Potassium ions.

Question 30

Given this F x I curve, is this neuron Type I or type II?

Answer 30

Type 1:

• (almost) arbitrarily low firing rates
• (mostly) linear increase of frequency with current

Question 31

What is plotted in the figure below?

Answer 31

An FxI curve

It measures the relationship between injected current and firing frequency of a given neuron.

Question 32

Given this F x I curve, what is the type of the neuron depicted?

Answer 32

A resonator,

A type 2 excitable neuron

• abrupt toggle between not firing and firing
• small modulation of frequency with current injected

Question 33

Define “Negative current”.

Answer 33

“Negative” current means that positive ions go into the cell and that the neuron depolarizes.

Question 34

The higher the capacitance, the _______ the potential difference comes to the resting potential.

Answer 34

the slower

Question 35

Why should neuroscientists learn nonlinear dynamical system theory?

Answer 35

Because neuron’s information-processing depends not only on the electrophysiological properties but also on their dynamical properties. Namely, the dynamics of bifurcation determine neuron’s excitability, spiking, its type, encoding ability, capacity for bistability, spontaneous activity and so on.

Question 36

What’s hyperpolarization?

Answer 36

When a neuron gets more negative.

Question 37

What’s depolarization?

Answer 37

When neurons get more positive.