Notes

Lecturer: Valerio Mante

Levels of Description Typical of biology to have multiple levels of description, and we don't know if each of them matter or not. Can we ignore some of these details and still get the same outcome when we try to replicate such a complex structure? No one can answer as of now.

Goal: create an artificial intelligent agent

• What is the essence of neural computations?
• What details can we ignore?
• What can we abstract?

Why study single neurons?

1. The computational unit of the brain constrains network computations.
2. Emulate in neuromorphic hardware.
3. Basis of experimental methods.

Neurons are Diverse Neurons are specialized to do some specific computations.

Two approaches to understanding neural computations:

• Biologically realistic simulations (Blue Brain Project).
• Simplify and Abstract.

Single Neuron Computations These kind of phenomena, such as transmission delays, dendritic computations and back-propagating action potentials, are not describable when the "point-neuron" model is used.

Neuromorphic Implementation

• Emulate biology
• Different substrate

Experimental Procedures

• Measure activity, e.g., calcium imaging.
• Perturb activity, e.g., optogenetic.

The Resting Potential Why do cells have membrane potential? It is a way to store energy. Neurons invest energy to perform concentration differences.

• Intracellular space, V = -70 mV.
• Extracellular space, V = 0 mV.

The Basic Ingredients

• The cell membrane (a barrier between inside and outside).
• Concentration gradients in vs. out (due to ionic pumps, they use energy to move ions in/out of the cell).
• Selective ionic channels (they are proteins, green channels in the pic, selective to specific ion types).

A thought Experiments The previous 3 ingredients give rise to action potentials. We are going to perform a thought experiment, the box to the left simulates a cell environment. At t = 0, the molecules are only inside the cell. At t = 0, the concentration inside is higher than outside. After some time t inf, the concentration reaches an equilibrium at a macroscopic level, even though at a microscopic level small exchanges continue to happen through the channels.

Now, we are going to assume more complex molecules, i.e., ions with a charge. Furthermore, the ionic channels are going to be selective, which allow only the passage of positive ions. What do we expect now at t inf? In this setting we do not reach the same equilibrium as before, indeed the inside starts to turn negatively charged every time a positive charge io goes outside the cell. But then, for every positive ion that goes out, the inside becomes more negative and attracts more the remaining positive ion channels, reducing the chances that further positive ions "escape".

The Cell Membrane This is essentially how things look like at the equilibrium.

Selective Ionic Channels & Ionic Flux There is asymmetry in the ionic flux, indeed, when one positive ion hits the channel from the outside, it will be dragged inside from the electric field. On the other side, a positive ion would need to have a kinetic energy bigger than $qV_{eq}$ to be able to cross the ionic channel from the inside.

The Boltzmann Factor The Boltzmann Factor is a statistical quantity that describes the probability of a system being in a certain energy state at thermal equilibrium. It is given by the following formula:

$$p(E_{kinetic} > \left| qV_{eq} \right|)\ = \ e^{\frac{qV_{eq}}{k_{B}T}} = \mathbf{\ }\mathbf{e}^{\frac{\mathbf{- E}}{\mathbf{k}_{\mathbf{B}}\mathbf{T}}}$$

where $E = Energy$, $\ k_{B} = Boltzmann\ constant\ = \ 1.38\ \times \ 10^{- 23}\ JK^{- 1}$ and $T = Temperature\ (K)$.

On the y-axis, $e^{\frac{qV_{\ }}{k_{B}T}}$, represents the percentage of the ions having enough energy to cross the ionic channel.

The Nernst Equation The Nernst Equation is an equation that describes the relationship between the electrical potential of a cell and the concentration of ions in the cell. It describes dependencies in equilibrium potential, which also take the name of reversal potentials

What about the assumptions? We assumed fixed concentration.

The Reversal Potential

Two Channel-Types The equilibrium is reached when the net current is 0.

Goldman-Hodgkin-Katz Equation

Energy Consumption in the brain