The simulations in Neuronify mimic neural networks in our body. This page is intended to give you a short introduction to basic concepts of neuroscience that you might encounter while using the app. In the future, this information may become part of Neuronify itself.

Brain tissue and cell types

Neocortex is the outer part of our brain. It is the big part of the brain that is underlying our skull. It is about 1-4 millimeters thick and consists of six layers. Each layer has some characteristic properties, but there are several connections between the layers.

In regular cortical tissue, there are two types of cells: Neurons and glia cells. Neuronify focuses solely on the neurons. Glia cells are traditionally known as helper cells, but recent research shows that they might be playing a bigger role in signal processing than previously thought.

The morphology of a neuron, i.e. its shape, is typically divded into three main parts: Dendrites, soma and axon. The dendrites receive input from other cells, which then travels through the cell body, known as the soma, before its projected out through the axon towards connections to other cells. These connections are known as synapses.

There are about 15-20 billion neurons in the human neocortex, with about 1000 - 10000 synapses per neuron. In total, that means there are about \(10^{14}\) connections between neurons in total. Neuronify is only capable of simulating a tiny fraction of these in about a thousand of real time speed. Other simulators can give valuable results from millions of neurons on a desktop computer. These simulators are typically not used for educational purposes alone, but rather for research on large-scale networks.

Electrical properties of neurons

The neuronal membrane is made of a lipid bilayer. This is made of two layers of lipid molecules. The molecules are phospholipids with a hydrophilic head and a hydrophobic tail. In water, the phospholipids arrange themselves so that the tails of one layer is directed towards the tails of the other layer. The hydrophilic heads are pointing towards the water. This bilayer structure allows almost no water and very few other molecules to enter and thus becomes a protective sheet that separates the interior of a cell from its environment.

Ion channels and pumps

Ion channels are proteins that are located in the cell membrane, passing through the lipid bilayer. Certain molecules are able to enter and leave the cells through these channels.

Some ion channels are voltage sensitive, meaning that they open or close depending on the voltage difference between the inside and outside of the cell. Others react with neurotransmitter molecules in the extracellular or intracellular fluids.

Another type of channels are simply passive. They are always open, but may still be selective to specific types of ions and molecules.

Ion pumps are energy consuming proteins that actively pump ions in and out of the cell. Unlike the ion channels, ion pumps are able to move ions against their concentration gradient. They are, among other things, responsible for maintaining the resting membrane potential.

Resting membrane potential

The potential difference over the membrane of a neuron is determined by the charge concentration on each side of the membrane. This is in turn depends on the flow of charge through the ion channels.

The resting membrane potential is defined as the membrane potential of a neuron at rest. The main mechanism behind the resting membrane potential comes from an initial concentration difference between the inside and outside of the cell membrane. For simplicity, let's assume that the cell exterior has a high sodium (Na+) and chloride (Cl-) concentration. The inside of the cell, on the other hand, has a high potassium (K+) and general anion (An-) concentration.

Let's assume that the cell membrane only has potassium ion channels open. This would allow only the potassium to leave the cell. Due to the concentration gradient, some potassium would leave the cell due to diffusion. However, as soon as the potassium starts to leave the cell, an electric gradient would start building up. When the force from the concentration gradient and the force from the electric field are equal, no more potassium would leave the cell. This would be an equilibrium, or resting, state.

The above mechanism is only one of the driving forces behind the resting membrane potential. Ion pumps and other ion channels are also involved in this process. Particularly important is the Na+-K+ pump that pumps sodium back out of cells and K+ into the cells. Considering only the above process of potassium leaving this cell, this may seem counter-intuitive, as we already see that potassium will leave the cell because of the concentration gradient. However, as we will see later, this is necessary to prepare the action potential - the main player in neuron signaling.

Electrogenic pumps are those that create a net movement of charge over the cell membrane.

At any given time, even during activity, the membrane potential is driven towards the resting membrane potential by the above mechanisms. This means that even after a long stretch of activity, the neuron will return to the resting membrane potential.

In neurons, the resting membrane potential is typically around -70 mV.

Nernst-Planck equation

To put numbers on the above ion movement through the ion channels, we need to take a closer look at the electrical force and the diffusion force. These both contribute to the total flux of particles through the cell membrane. This flux is given by the Nernst-Planck equation, which we will give a simplified derivation of in this section.

Electrical drift

Charged particles are subject to an electric force when in an electric field:

\[ \mathbf F_E = q \mathbf E \]

We will change the notation slightly, because it's customary to write charge with a big \(Z\) in chemistry:

\[ \mathbf F_E = Z \mathbf E \]

The electric field is the gradient of the potential:

\[ \mathbf F_E = Z \nabla V \]

With no other external forces, the mean drift of a charged particle is equal to this force:

\[ \mathbf F_E = \frac{1}{\mu} \mathbf v_\mathrm{drift} \]

Here \( \mu \) is the friction from collision with other particles. The two expression for \( F_E \) may be joined to one equation:

\[ Z \nabla V = \frac{1}{\mu} \mathbf v_\mathrm{drift} \]

The Einstein relation is

\[ \mu = \frac{D}{k_B T}, \] where \( k_B \) is Boltzmann's constant, \( T \) is the temperature and \( D \) is the diffusion constant for the given species.

We then get the drift velocity as

\[ \mathbf v_\mathrm{drift} = \frac{Z \nabla V D}{k_B T}. \]

The flux is the velocity times the number concentration (number of particles per volume) \( c \):

\[ \mathbf j_\mathrm{drift} = \frac{Z c D}{k_B T} \nabla V. \]

This is the electric contribution to the flux.

Diffusion

The diffusive flux is given by Fick's law:

\[ \mathbf j_\mathrm{diff} = - D \nabla c. \]

For a derivation of this, see for instance the Wikipedia article on Fick's law.

Concentrations in moles

It is customary to write the flux in terms of concentrations in moles, rather than regular numbers. Further, we replace Boltzmann's constant with the gas constant and Faraday's constant:

\[ \mathbf J_\mathrm{drift} = \frac{Z F D}{R T} [X] \nabla V. \] \[ \mathbf J_\mathrm{diff} = - D \nabla [X]. \]

Here \( [X] \) is the concentration of species \( X \) in moles.

Total flux

If we sum the two contributions to the flux, we get the Nernst-Planck equation:

\begin{align} \mathbf J &= J_\mathrm{diff} + \mathbf J_\mathrm{drift} \\ &= - D \left ( \nabla [\mathrm{X}] - \frac{Z F}{R T} [\mathrm{X}] \nabla V \right ) \end{align}

Nernst potential

We assume that only one species is free to move so that we can treat all other as a static background electric field. The Nernst potential is then given when there is no net flux of the given species. This can be found by setting the Nernst-Planck equation to zero, i.e. no net flux. Further, we focus only on the potential difference in one direction, the \( x \)-direction.

\begin{align} \int_{x_1}^{x_2} \frac{1}{[\mathrm X]} \frac{d[\mathrm X]}{dx} \, dx &= \frac{Z F}{R T} \int_{x_1}^{x_2} \frac{dV}{dx} \, dx \\ \int_{[\mathrm X]_1}^{[\mathrm X]_2} \frac{1}{[\mathrm X]} d[\mathrm X] &= \frac{Z F}{R T} \int_{V_1}^{V_2} dV \\ \ln \frac{[\mathrm X]_2}{[\mathrm X]_1} &= \frac{Z F}{R T} (V_2 - V_1) \\ E_{\mathrm X} &= \frac{R T}{Z F} \ln \frac{[\mathrm X]_2}{[\mathrm X]_1}. \end{align} The potential difference \( E_{\mathrm X} = V_2 - V_1 \) is known as the Nernst potential for the species \( \mathrm X \).

Goldman-Hodgkin-Katz equations

When multiple ion species are involved, we need to expand on the formalism outlined above. We assume that the ions cross the membrane independently and that the electric field within the membrane is constant. As above, the movement of the ions is governed by the electric field and the concentration gradient across the membrane. For each ion, we have the flux as

\begin{align} \mathbf J_\mathrm{X} &= J_\mathrm{X,diff} + \mathbf J_\mathrm{X,drift} \\ &= - D \left ( \frac{d [\mathrm{X}]}{dx} - \frac{Z F}{R T} [X] \frac{dV}{dx} \right ) \end{align}

From our assumption of a constant electric field inside the membrane, we get

\begin{align} \mathbf J_\mathrm{X} &= J_\mathrm{X,diff} + \mathbf J_\mathrm{X,drift} \\ &= - D \left ( \frac{d [\mathrm{X}]}{dx} - \frac{Z F}{R T} [X] \frac{E_\mathrm{m}}{L} \right ) \end{align} where \(E_\mathrm{m}\) is the membrane potential and \(L\) is the membrane thickness.

By separation of variables, we obtain

\begin{align} dx &= \frac{d[\mathrm X]} {-\frac{J_\mathrm{X}}{D_X} + \frac{D_\mathrm{X} Z_\mathrm{X} F}{RT} [\mathrm{X}] \frac{E_\mathrm{m}}{L}}. \end{align}

Integration on both sides from \(x\) inside the membrane to outside the membrane gives the solution

\begin{align} J_\mathrm{X} = P_\mathrm{X} Z_\mathrm{X} \mu \left ( \frac{[X]_\mathrm{out} - e^{Z_\mathrm{X}\mu} [X]_{\mathrm{in}} } {1 - e^{\mu Z_\mathrm{X}}} \right), \end{align}

where

\[ \mu = \frac{FE_\mathrm{m}}{RT}. \]

and

\[ P_\mathrm{X} = \frac{D_\mathrm{X}}{L}. \]

is the permeability of the membrane to ion \(\mathrm{X}\).

This equation may now be solved for the membrane potential \(E_\mathrm{m}\). How to solve it depends on the valency of the ions involved. In the simplest case, where all involved ions are monovalent so that \(Z_\mathrm{X} = \pm 1 \), the solution becomes

\[ E_\mathrm{m} = \frac{RT}{F} \ln \frac{\sum_\mathrm{cations C} P_\mathrm{C} [\mathrm{C}^+]_\mathrm{out} + \sum_\mathrm{anions A} P_\mathrm{A}[\mathrm{A}^-]_\mathrm{in}} {\sum_\mathrm{cations C} P_\mathrm{C} [\mathrm{C}^+]_\mathrm{in} + \sum_\mathrm{anions A} P_\mathrm{A}[\mathrm{A}^-]_\mathrm{out}} \]

Inserting the concentrations and permabilites into this equation yields the equilibrium membrane potential.

Note that the notion of the permeability predates the discovery of ion channels. The permeability is therefore dependent on the number of ion channels that are open and may vary with time in active membranes.

Neuron signaling

Within a neuron, a propagating signal is known as an action potential. The action potential is driven by ion channels that allow different charges to flow through.

A synapse may be chemical or electrical, depending on its function. In a chemical synapse, the presynaptic neuron (sender) emits neurotransmitters (molecules) that activate reactive sites on the postsynaptic (receiver) neuron. In an electrical synapse, the two neurons are interconnected with shared ion channels. This keeps the contents of the intracellular fluid of the two neurons in sync. Functionally, the biggest difference between electrical and chemical synapses is in speed and amplification. Chemical synapses can amplify signals, while electrical synapses are much faster. In Neuronify we deal mainly with chemical synapses. However, note that current clamps and similar items inject current directly into a receiving neuron, rather than to trigger its synapse.

Neurons may be either excitatory or inhibitory. Excitatory neurons have synapses that increase the membrane potential of the postsynaptic neuron. Inhibitory neurons decrease the membrane potential. About 80 % of the cortical neurons are excitatory, while 20 % of the neurons are inhibitory.