Modeling Neuron Electrokinetics using Markov Models

By Eric Cruet

Ion Channel Kinetics

In continuation of the previous post, the function of neurons in the brain is about detection.  They receive thousands of different input signals from other neurons, trained to detect patterns specific to their function.  A simplistic analogy is the thermostat in an oven.  When you set the oven to preheat at 350 degrees, the sensor samples the temperature until it reaches the specified threshold temperature.  Then it “fires” a signal and the alarm goes off.  In the same fashion, the neuron has a threshold and “fires” a signal to adjacent neurons only if it detects a signal significant enough to cross its threshold.  This signal is known as the action potential or spike and in the diagram above, is represented and accomplished by the excitability arrows.

Synapses are the connectors between sending neurons, dendrites are the “branches” that integrate all the inputs to the neurons, and the part of the axon that’s very close to the output end of the neuron (Axon hillock) is where the threshold activity takes place.  The farthest end of the axon branches out and turns into inputs to other neurons, completing the next chain of communication.  See the image below.

Neuron Cell Structure

The bottom line is understanding the neuron’s fundamental functionality as a detection mechanism: it receives and integrates inputs, and determines whether its threshold has been exceeded, triggering an output signal based on its inputs.

Now let’s briefly cover some basic biochemistry, since Markov models simplify and simulate ion channel kinetics.  Ion channels are where the some of vital functions involved in the triggering of the signal occur.

There are three major sources of input signals to the neuron [4]:

    1. Excitatory inputs:  these are the more common, prevalent type of input from other neurons (=approx 85% of all inputs).  Their effect excites the receiving neuron, which makes it more likely to exceed its threshold and “fire”, or trigger a signal.  These are signalled via a synaptic channel called AMPA, opened by the neurotransmitter glutamate.  In addition, AMPA receptors that are non-selective cationic channels allowing the passage of Na+ and K+and therefore have an electric equilibrium potential near 0 mV (milliVolts).
    2. Inhibitory inputs: comprising the other 15% of inputs, they have the opposite effect of the excitatory inputs.  They cause the neuron to be less likely to fire, or trigger a signal, which makes the integration process (of inputs) much more robust (by keeping the excitation under control).  Specialized neurons in the brain called inhibitory interneurons accomplish this function in the brain.  These inputs are signalled via GABA (gamma-Aminobutryc Acid) synaptic channels, via the GABA neurotransmitter.  It also causes the opening of ion channels to allow the flow of either negatively charged Cl (chloride) ions into the cell or positively charged K+(potassium) ions out of the cell.
    3. Leak inputs: technically not considered inputs since they are always active.  However, they are similar to inhibitory inputs in that they counteract excitation and keep the neuron in balance.  They receive their signalling via K+ (potassium) channels.  

The interaction between these elements in a cell create what is know as the membrane potential.  Membrane potential (also transmembrane potential or membrane voltage) is the difference in electrical potential between the interior and the exterior of a biological cell. Typical values of membrane potential range from –40 mV to –80 mV.  These are a result of differences in concentration of ions ( Na+/K+/Cl) on opposite sides of a cellular membrane.  Please refer to the following picture:

 

So we’ve covered the process by which the neuron “detects” various inputs based on chemistry.  These chemical processes generate a difference in potential (charge or mVolts) across the cell.  In simplified terms, the rate, direction, and the amount of change in this potential is what determines whether a neuron will exceed its threshold.  A brief overview of mathematical models for neuron ion channel kinetics follows.

Hodgkin-Huxley

The first, most widely-used models of neurons that is based on the Markov kinetic model was developed from Hodgkin and Huxley’s 1952 work [2] based on data from the squid giant axon. We note as before our voltage-current relationship, this time series generalized to include multiple voltage-dependent currents:

C_\mathrm{m} \frac{d V(t)}{d t} = -\sum_i I_i (t, V).

Each current is given by Ohm’s Law as: (this is derived from the basic  I = \frac{V}{R},   ) where

I = Current, V = Voltage and R = Resistance or 1/g where g = Conductance

I(t,V) = g(t,V)\cdot(V-V_\mathrm{eq})

where g(t,V) is the conductance over time, or inverse resistance, which can be expanded in terms of its constant average  and the activation and inactivation fractions m and h, respectively, that determine how many ions can flow through available membrane channels. This expansion is given by

g(t,V)=\bar{g}\cdot m(t,V)^p \cdot h(t,V)^q

and our fractions follow the first-order kinetics

\frac{d m(t,V)}{d t} = \frac{m_\infty(V)-m(t,V)}{\tau_\mathrm{m} (V)} = \alpha_\mathrm{m} (V)\cdot(1-m) - \beta_\mathrm{m} (V)\cdot m

with similar dynamics for h, where we can use either τ and m or α and β to define our gate fractions.

With such a form, all that remains is to individually investigate each current one wants to include. Typically, these include inward Ca2+ and Na+ input currents and several varieties of K+ outward currents, including a “leak” current. The end result can be at the small end 20 parameters which one must estimate or measure for an accurate model [1].  At that time this could not be computed.  This was the starting point for a subsequent series of studies, all attempting to simplify the neuron model. 

In 2008, James P. Keener, performing research in mathematics at the University of Utah, published a paper entitled “Invariant Manifold Reductions for Markovian Ion Channel Dynamics.”  He proposed using Markov jump processes to model the transitions in ion channel states [3]. These Markov models had been previously been used in conductance based models to study the dynamics of electrical activity in nerve cells, cardiac cells and muscle cells.

In summary, what Dr. Keener proved was that the classical Hodgkin-Huxley formulations of potassium and sodium channel conductance are exact solutions of Markov models, although there were no means of computing proof at the time. This means that the solutions of the Hodgkin-Huxley equations and the solutions of a full Markov model simulating neuron electrokinetic activity with an 8-state sodium channel and a 4-state potassium channel model (after several milliseconds during which initial transients decay) are exactly the same, even though the first is a system of four differential equations and the latter is a system of 13 differential equations.

There are a lot of pieces to the cognitive neuroscience puzzle.  This is one of many theoretical frameworks to approach the complex subject of brain function.  One of the drawbacks of a computational cognitive approach is that it basically designs the very functionality it tries to explain.  It is still very useful, but limited in what it can ultimately explain. It has also been one of the most successful in dealing with cognitive function precisely because it deals at a higher level modeling a system that uses mathematics and logic with the same tools.  The ultimate goal is that it piques the curiosity of those who are unfamiliar to the subject and share what I’ve learned with those who have acquired an interest.

“The larger the island of knowledge, the longer the shoreline of wonder.”

Ralph Washington Sockman (1889 – 1970)

 

References:

 

[1] Goldwyn, J. H., & Shea-Brown, E. (2011). The what and where of adding channel noise to the Hodgkin-Huxley equations. PLoS computational biology7(11), e1002247.
[2] Hodgkin, A. L., & Huxley, A. F. (1952). Propagation of electrical signals along giant nerve fibres. Proceedings of the Royal Society of London. Series B, Biological Sciences140(899), 177-183.
[3] Keener, J. P. (2009). Invariant manifold reductions for Markovian ion channel dynamics. Journal of mathematical biology58(3), 447-457.
[4] O'Reilly, R. C., Munakata, Y., Frank, M. J., & Hazy, T. E. (2012). Computational Cognitive Neuroscience. Wiki Book,.