Basic computational models in neuroscience

Question 1

Why make computational models? Isn’t neuroscience complicated enough already?

Answer 1

The goal of computational models is to make our life easier: we can easily test a hypothesis in a system that is way less complicated than the real brain.

Question 2

How does making computational models make our lives easier

Answer 2

We can model crucial elements to make simple predictions about process. If our hypothesis has a red flag, we’ll see it quickly without having to go all the way in the training/ surgery/ recording/ analysis cycle. They can also help us to spot things that we missed from the data, and refine our experimental design.

Question 3

Why is learning to model a good idea even if you want to be a ‘pure’ researcher?

Answer 3

Experiment and model are already combined in most cutting-edge research. Even if you plan to be a ‘pure’ experimentalist, knowing about models will make your life much easier when you decide to collaborate with a modeller.

Question 4

Describe the research Jorge described as a practical example of modelling

Answer 4

A recent experimental study in our group showed, using ferrets, that different (low/high) levels of awareness lead to different oscillatory patterns in several brain areas. Changes in awareness may lead to modulations of visual and parietal cortices from frontal cortex, but technical limitations prevented us to precisely identify, in our experiment, the circuits responsible for these effects.

For this reason, we built a computational model to reproduce the patterns of oscillatory brain activity. Among other things, the model predicted which anatomical connections (from the hundreds of possible options) may produce such activity patterns.

Question 5

Are computational and theoretical models suited for more molecular process or systems processes?

Answer 5

Computational and theoretical models can be used to describe neural systems at many different scales –from biophysical processes inside synapses to full brain dynamics. They also embrace different approaches, from highly detailed biophysical descriptions to more abstract models.

Question 6

What could be inferred from differences in the model predictions and the observed data?

Answer 6

Whatever the shape and colour, computational models can use existing knowledge and data to deliver predictions, which then can be used in the design of present and future experiments.

Differences in model predictions could suggest additional mechanisms which are not accounted for in the model. Parameter values and interactions can then be tweaked to become closer to the predicted data.

Question 7

What else can be derived from models, closely related to predictions?

Answer 7

Models can also be used to make postdictions (Abbott, 2008): to explain known phenomena from a different perspective, or to integrate previous findings into a larger, coherent formal framework. Postdictions can tell you if your assumptions are sufficient

Question 8

From a more methodological point of view, computational models in neuroscience come from different mathematical branches. Name these (3)

Answer 8

1) The statistical approach (for example, Bayesian neural models)

2) The computer science approach (such as artificial neural networks, deep learning…);while artificial neural networks are not the same, they can behave like real networks

3) The biophysical approach (such as biophysical models of neurons, tools coming from physics and complex systems science)

Question 9

Which of these three mathematical branches are we most concerned with? What is it most concerned with?

Answer 9

Biophysical; based on what actually happens in the model. The level of detail is up to you

Question 10

Membrane potential arises from a separation of positive and negative charges across the cell membrane. Where do these excess charges largely stem from?

Answer 10

Excess charges are concentrated directly at the membrane (in- and outside). Each excess is only a tiny fraction ot the total number of ions in/outside the cell

Question 11

What is the use of having a membrane potential? (high energy demand)

Answer 11

• rapid electrical signalling possible; rapid transport of signals through CNS

• You only have to open “the gates” to use ion gradients to change the voltage

Question 12

How do you calculate the membrane potential?

Answer 12

Vm= V in– V out
intracellular voltage - extracellular voltage

Question 13

What is meant by the resting potential and what is the typical voltage?

Answer 13

•Resting membrane potential: “undisturbed” state
•Usually RMP ~ -85 to -60 mV. Thus, excess negative charge inside cell at rest

Question 14

What is meant by polarisation?

Answer 14

Any potential difference V in– V out that is different from zero. In practice: Vm is negative

Question 15

What is meant by de-polarisation

Answer 15

loss of (negative) polarization, so Vm moves towards 0 mV

Question 16

What is meant by hyper-polarisation?

Answer 16

Extra (negative) polarisation, so Vm moves away from 0 mV, becomes more negative than rest

Question 17

What is meant by the direction of current flow? I.e if ions were moving in and out of the cell, which direction would the current be flowing?

Answer 17

Direction of current flow is defined as direction of net movement of positive charge e.g. “inward current”: net movement of + charge into cell (or: negative charge leaves cell to outside!)

Question 18

Name and describe the two driving forces at the membrane

Answer 18

Electrical driving force: depends on membrane potential (+ions attracted to inside)

Chemical driving force: depends on concentration difference of ion across membrane

Question 19

What is meant by an equilibrium potential?

Answer 19

For a given molecule, what the membrane potential must be at for them to be at their concentration potential. Orin other words:

Equilibrium potential (=reversal potential): the potential at which the electrical driving force equals the chemical (diffusional) driving force

Question 20

Give the equilibrium potentials for the four main ions involved in the whole transmembrane community

Answer 20

ion | Inside Mm | Outside Mm | Eq pot

Na+ 50| 440| +55mV|
Cl - 52| 560| -60mV|
K + 400| 20| -75mV|
Organic ions (A-) 385| ~0| –|
Ca 2+ 0.0001| 2| +120 mV|

Question 21

What does the Nernst equation tell us?

Answer 21

It gives the equilibrium potential for each ion

Question 22

Give the Nernst equation and what each variable represents

Answer 22

Ek = ( RT / zF ) ( ln [K+]out / K+ in )

R: gas constant
T: temperature (Kelvin)
z: ion valence
F: Faraday constant

RT/F ~ 25 mV at 25 oC

Question 23

What is meant by the faraday constant and what is it for potassium?

Answer 23

In physical chemistry, the Faraday constant, denoted by the symbol F, is the electric charge per mole of elementary charges. It represents the amount of electric charge carried by one mole, or Avogadro’s number, of electrons.

Question 24

What do we assume in the Nernst equation? (2)

Answer 24

We assume that pumps establish a gradient in the 1st place, and that membrane is somewhat permeable to K+ ions (for calculating K+)

Question 25

Describe the situation with multiple ion species (Na+ and K+) from rest to modest depolarisation and further

Answer 25

•At rest, electrical and chemical equilibrium is maintained.
•The opening of Na+ channels (stimulus) drives Na+ ions inside the cell, depolarising the cell membrane.
• As a reaction to this depolarization, K+ electrical forces become weaker and K+ flows out of the cell, restoring balance.

Question 26

If concentration gradient of K+ is present, when does equilibrium (i = 0) arise?

Answer 26

If concentration gradient of K+ is present, equilibrium (i=0) only arises if potential is negative

Question 27

What is the Goldman-Hodgkin-Katz equation used for and how does it do it?

Answer 27

Membrane potential is determined by a weighted sum
of equilibrium potentials, where weight is determined
by the relative permeability of the ion

Question 28

Give the Goldman-Hodgkin-Katz equation and name the variables

Answer 28

Vm=R.T / F ln( ( Pk .[K+]out + Pcl . [Cl-]in ) / ( Pk .[K+]in + Pcl . [Cl-]in ) )

R: gas constant
T: temperature (Kelvin)
F: Faraday constant
Pk: Permeability of K

Question 29

Why is temperature important in these models

Answer 29

It is indicative of the amount of movement (Kinetic energy?) of the ions and therefore the diffusion process

Question 30

How is the electrical driving force represented in neural depictions as circuits?

Answer 30

electrical driving force represented by “battery” symbol.

Question 31

How is current-voltage relation calculated/ represented mathematically?

Answer 31

Through Ohm’s law:
Ik= γ*(Vm-Ek)

Where Ik is the current generated by an ion equation; shows the relationship between this and the membrane

Question 32

How would you describe the relationship between current and voltage?

Answer 32

Current-voltage plot is linear (ohmic)

and i=0 if V=0 in absence of ionic gradient

Question 33

How are ionic channel groups represented as electric circuits?

Answer 33

Several K+channels situated in parallel: lump into one
conductance. conductance placed in series with battery (=electromotive force). Different types of channels in parallel: different equilibrium potentials => different battery-conductance combinations.

Question 34

What is the Hodgkin-Huxley model meant to model? How was this model formulated (methods)?

Answer 34

Electrophysiological (voltage clamp) recordings of the giant axon of the squid. These recordings were
followed by a computational model of the ionic contributions to the membrane potential. The model provided a biophysical description of the generation of action potentials (APs).

Question 35

What are the Hodgkin-Huxley model assumptions?

Answer 35

• Two types of ion channels (Na+ and K+) regulate the membrane potential dynamics of the cell.

• “Active membrane”: the ion channel conductance varies as a function of time and voltage.

Question 36

What resisitences did Hodgkin and Huxley consider for their model and how accurate were they?

Answer 36

They considered three different resistances in their model; Na, K and C, passive resistances. They assumed chloride had a significant role in generating individual action potentials. While it does play a role it is not a fundamental part as suggested. You can test your model to reveal things like this

Question 37

What does C represent in these circuitry models?

Answer 37

Capacity: How much energy the neuron is able to store

Question 38

When is an electric current generated in the neuron?

Answer 38

If the membrane potential is different to the charge of an ion then a current is generated. If not they are satisfied

Question 39

Describe two different ion channels as described by the hodgkin huxley model

Answer 39

Potassium channel:
• The K+ channel has four similar subunits
• Each subunit can open or close independently.
• The channel is open if (and only if) all four subunits are open.

Sodium channel:
• The Na+ channel has three “fast” subunits and one “slow” subunit.
• Each subunit can open or close independently.
• The channel opens if all four subunits are open. In practice, the slow subunit works as an inactivation mechanism due to its particular “inverse” dynamics. This means they tend to open and close at opposite times but small windows of opportunity.

Question 40

Break down the Hodgkin-Huxley model without the inclusion of ion channel dynamics

Answer 40

the extracellular current (Iext) is the sum of the other currents (Ina + Ik + IL + Ic). The current is given by the ion conductance (g) multiplied by the membrane potential (difference in voltages; Vx - Vna ). E.g the sodium current is given by (Ina = gna (V- Vna )). The current of the capacitor is calculated by the derivative of the capcitor (C dv/dt). These are then summed to get the model:

Iext = gna (V- Vna ) + gK (V- VK ) + gL (V- VL ) + C dv/dt

Which can be rewritten as

C dv/dt = gna (Vna- V ) + gna (Vk- V ) + gna (VL- V ) + Iext

Question 41

Break down the inclusion of the ion channel dynamics in the model

Answer 41

If we consider:
• The probability of one subunit being open as 𝒏
• The probability that the channel is open as 𝒏^4
• The maximal K+ conductance for all channels open as
𝒈𝑲
Then the K+ conductance is given by:
𝒈𝑲 . 𝒏^4 (Vk - V )

If we consider:
• The probability of a fast subunit being open as 𝒎
• The probability of a slow subunit being open as 𝒉(controls inactivation)
• The probability that the channel is open as 𝒎𝟑𝒉
• The maximal Na+ conductance for all channels open as “𝒈𝑵𝒂
Then the Na+ conductance is given by:
𝒈𝑵𝒂 . m^3 . h (V𝑵𝒂 - V )

So the equation becomes:
C dv/dt = 𝒈𝑵𝒂 . m^3 . h (V𝑵𝒂 - V ) + 𝒈𝑲 . 𝒏^4 (Vk - V ) + gna (VL- V ) + Iext

Question 42

How are the dynamics of the K+ and Na + subunits given?

Answer 42

Tn dn/dt = -n + n∞ (activation of potassium)
Tm dm/dt = -m + m∞ (activation of sodium)
Th dh/dt = -h+ h∞ (inactivation of sodium)

Here, the variables 𝜏&,(,)and [𝑛,𝑚,ℎ]*depend on the membrane potential V (and therefore, also vary on time with V).

Question 43

What do these subunit equations interact to give?

Answer 43

These equations interact to generate action potentials in the model i.e the depolarisation peak and the after-spike hyper-polarisation (see in docs)

Question 44

Give the time course of an AP according to the HH model

Answer 44

1) Initial depolarisation needed
2) Na+channels open
3) They depolarise the membrane further (not in V-clamp)
4) More Na+channels open => more depolarisation (positive feedback loop!)
5) K+channels start to open: repolarising effect
6) Na+channels inactivate
7) K+conductance outlasts Na+ conductance: hyperpolarisation

Question 45

Give four examples of how different types of ion channels can have a variety of cell firing behaviours

Answer 45

(A) N. tractus solitarius: hyperpolarize cell tonically => depolarising pulse triggers spikes, but delayed
due to A-current (carried by potassium ions)

(B) Thalamic neurons: hyperpolarise cell tonically => depolarising pulse triggers burst of spikes,
due to low-threshold Ca-current

(C) Thalamocortical relay neuron: spontaneous burst of action potentials, due to:
1) H-current, switches on during hyperpolarisation but acts to depolarise cell;
2) low-threshold Ca-current, triggering Na-action potentials
3) repolarisation due to shut-down of H- and Ca-current, versus resting K+ currents

D) Neurotransmitter ACh closes M-current channels in sympathetic ganglion (ANS)
M-current is a K+current that slowly opens upon depolarisation;
cell can fire only one spike when depolarised; ACh => closes M-current => more spikes