Compartmental neuron models

• Compartmental modelling is a very natural way of modelling dynamical systems that have certain inherent properties with conservation principles. The compartmental modelling is an elegant way, a state space formulation to elegantly capture the dynamical systems that are governed by the conservation laws. Whether it is the conservation of mass, energy, fluid flow or information flow. Basically, they are models whose state variables tend to be non-negative (such as mass, concentrations, energy). So the equations for mass balance, energy, concentration or fluid flow can be written. It ultimately goes down to networks in which the brain is the largest of them all, just like Avogadro number, very large amount of molecules that are interconnected. The brain has very interesting interconnections. On a microscopic level thermodynamics is virtually impossible to understand but from a macroscopic view we see that these follow some universal laws. In the same way brain has numerous interconnections, which is almost impossible to write a differential equation for. (These words were said in an interview by Dr. Wassim Haddad )
• General observations about how the brain functions can be made by looking at the first and second thermodynamic laws, which are universal laws. Brain is a very large-scale interconnected system; the neurons have to somehow behave like the chemical reaction system, so, it has to somehow obey the chemical thermodynamic laws. This approach may lead to more generalized model of brain. (These words were said in an interview by Dr. Wassim Haddad )

a) Branched dendrites viewed as cylinders for modelling. b) Simple model with three compartments

• Complicated dendritic structures can be treated as multiple compartments interconnected. The dendrites are divided into small compartments and they are linked together as shown in the figure.[1]
• It is assumed that the compartment is isopotential and spatially uniform in its properties. Membrane non-uniformity such as diameter changes, and voltage differences are occurred in between the compartments but not inside them.[1]
• An example of a simple two-compartment model:

Consider a two-compartmental model with the compartments viewed as isopotential cylinders with radius ${\displaystyle a_{i}}$ and length ${\displaystyle L_{i}}$.

${\displaystyle V_{i}}$ is the membrane potential of ith compartment.

${\displaystyle c_{i}}$ is the specific membrane capacitance.

${\displaystyle r_{Mi}}$ is the specific membrane resistivity.

The total electrode current, assuming that the compartment has it, is given by ${\displaystyle I_{\text{electrode}}^{i}}$.

The longitudinal resistance is given by ${\displaystyle r_{L}}$.

Now according to the balance that should exist for each compartment, we can say

${\displaystyle i_{\text{cap}}^{i}+i_{\text{ion}}^{i}=i_{\text{long}}^{i}+i_{\text{electrode}}^{i}}$.....eq(1)

where ${\displaystyle i_{\text{cap}}^{i}}$ and ${\displaystyle i_{\text{ion}}^{i}}$ are the capacitive and ionic currents per unit area of ith compartment membrane. i.e. they can be given by

${\displaystyle i_{\text{cap}}^{i}=c_{i}{\frac {dV_{i}}{dt}}}$ and ${\displaystyle i_{\text{ion}}^{i}={\frac {V_{i}}{r_{Mi}}}}$.....eq(2)

If we assume the resting potential is 0. Then to compute ${\displaystyle i_{\text{long}}^{i}}$, we need total axial resistance. As the compartments are simply cylinders we can say

${\displaystyle R_{\text{long}}={\frac {r_{L}L_{1}}{2\pi a_{1}^{2}}}+{\frac {r_{L}L_{2}}{2\pi a_{2}^{2}}}}$.....eq(3)

Using ohms law we can express current from ith to jth compartment as

${\displaystyle i_{\text{long}}^{1}=g_{1,2}(V_{2}-V_{1})}$ and ${\displaystyle i_{\text{long}}^{2}=g_{2,1}(V_{1}-V_{2})}$ .....eq(4)

The coupling terms ${\displaystyle g_{1,2}}$ and ${\displaystyle g_{2,1}}$ are obtained by inverting eq(3) and dividing by surface area of interest.

So we get

${\displaystyle g_{1,2}={\frac {a_{1}a_{2}^{2}}{r_{L}L_{1}(a_{2}^{2}L_{1}+a_{1}^{2}L_{2})}}}$

and

${\displaystyle g_{2,1}={\frac {a_{2}a_{1}^{2}}{r_{L}L_{1}(a_{1}^{2}L_{2}+a_{2}^{2}L_{1})}}}$

Finally, ${\displaystyle i_{\text{electrode}}^{I}={\frac {I_{\text{electrode}}^{i}}{A_{i}}}}$

${\displaystyle A_{i}=2\pi a_{i}L_{i}}$ is the surface area of the compartment i.

If we put all these together we get

${\displaystyle c_{1}{\frac {dV_{1}}{dt}}+{\frac {V_{1}}{r_{M1}}}=g_{1,2}(V_{2}-V_{1})+{\frac {I_{\text{electrode}}^{1}}{A_{1}}}}$

${\displaystyle c_{2}{\frac {dV_{2}}{dt}}+{\frac {V_{2}}{r_{M2}}}=g_{2,1}(V_{1}-V_{2})+{\frac {I_{\text{electrode}}^{2}}{A_{2}}}}$.....eq(5)

If we use ${\displaystyle r_{1}=1/g_{1,2}}$ and ${\displaystyle r_{2}=1/g_{2,1}}$ then eq(5) will become

${\displaystyle c_{1}{\frac {dV_{1}}{dt}}+{\frac {V_{1}}{r_{M1}}}={\frac {V_{2}-V_{1}}{r_{1}}}+{\frac {I_{\text{electrode}}^{1}}{A_{1}}}}$

${\displaystyle c_{2}{\frac {dV_{2}}{dt}}+{\frac {V_{2}}{r_{M2}}}={\frac {V_{1}-V_{2}}{r_{2}}}+{\frac {I_{\text{electrode}}^{2}}{A_{2}}}}$.....eq(6)

Now if we inject current in cell 1 only and each cylinder is identical then ${\displaystyle r_{1}=r_{2}\equiv r}$

Without loss in generality we can define ${\displaystyle r_{M}=r_{M1}=r_{M2}}$

After some algebra we can show that

${\displaystyle {\frac {V_{1}}{i_{1}}}={\frac {r_{M}(r+r_{M})}{r+2r_{M}}}}$

also

${\displaystyle {\frac {R_{\text{input,coupled}}}{R_{\text{input,uncoupled}}}}=1-{\frac {r_{M}}{r+2r_{M}}}}$

i.e. the input resistance decreases. For increment in the potential, coupled system current should be greater than that is required for uncoupled system. This is because the second compartment drains some current.

Now, we can get a general compartmental model for a treelike structure and the equations are

${\displaystyle C_{j}{\frac {dV_{j}}{dt}}=-{\frac {V_{j}}{R_{j}}}+\sum _{k{\text{ connected }}j}{}{\frac {V_{k}-V_{j}}{R_{jk}}}+I_{j}}$ [1]

Dendrites and axons are considered to be continuous (cable-like), rather than series of compartments.[1]

• Computational neuroscience
• Dynamical system
• Multi-compartment model
• Connectionism
• Neural network
• Biological neuron models
• Neural coding
• Brain-computer interface
• Neural engineering
• Neuroinformatics
• Mathematical model
• Compartmental models in epidemiology
• Physiologically-based pharmacokinetic modelling

• : Ted talk on supercomputing Archived 2014-02-11 at the Wayback Machine
• : Laboratory of Biological Modelling
• : Pharmacokinetics Modelling Tutorial
• : Usefulness of two compartmental model in Pharmacokinetics
• : Pharmacokinetics