1. Biological Neural Networks
Neural Computation
Biological Neural Networks

2. Sources
Material in this lecture comes from, “Handbook of Natural Computing,” Editors Grzegorz Rosenberg, Thomas Back and Joost N. Kok, Springer 2014.

3. Computational neuroscience studies the nervous system from the point of view of its functionality and relies both on experimental data and on theoretical models of individual neurons and networks
Single neuron descriptions/models
Signal processing through synapses
Small circuits
Large neural networks
How detailed does the description of a neuron have to be to build a model that is biologically realistic and computationally tractable?

4. A living neuron maintains a voltage drop across its membrane
Define the voltage outside the cell as zero
At rest the inside of the cell is about -70mV (+/- 20mV)
Due to difference in ion concentrations
4 different charge carriers: K+ , Cl- , Na+ , Ca2+
If cell membrane were permeable to Na it would flow in:
Due to concentration gradient
Because of the attraction of the negative membrane to the positive Na
Influx of Na only stops if the voltage across the membrane is +55mV, the reversal potential of Na
Nernst equation (z is the valency of the ion):
At rest the Na channels are largely closed and only very little Na can flow in
The K and Cl channels are somewhat open yielding a rest potential of -70mV
No net current flow; concentration gradient of ions is actively maintained with ion-pumps and exchangers (these proteins move ions across the membrane at the expense of energy)

6. Model passive neuron model (at resting voltage) using resistors and capacitors of a single compartment cell:
Injected current is Iinj, total capacitance is C, total membrane resistance is R
Current through resistor:
Current flows until capacitor is charged up:

7. Time constant is independent of cell area:
Capacity is proportional to the membrane area A (the bigger the more charge it can store)
Resistance is inversely proportional to A: define resistivity r_m=AR_m, conductance g=1/R
Model is correct for small perturbations around the resting potential

8. More realistic is a cable model where many cylindrical compartments are coupled to each other with resistors
Resistance between cables is called axial resistance
Diameter of the cylinder is d, length h
Resistance between compartments is 4ri*h/(pi*d2), where ri is the intracellular resistivity
Surface area cylinder compartment is pi*h*d  membrane resistance rm/(pi*h*d) and cylinder capacitance cm*pi*h*d

11. Fast events like spikes: time-constant neuron should be reduced
Reduce membrane capacitance  biophysically impossible
Dramatically increase conductance through the membrane  this is the basis for spike generation

12. Open-probability of a channel depends on the voltage across the membrane
Action potential initiates:
Close to the threshold voltage a few Na channels start to open
[Na] higher outside the cell (because its reversal potential is +40mV)  inflow Na, depolarizing cell
Opens even more Na channels and the spike initiates
Rapidly after spike starts, Na channels close again, K channels open
Outflow K, hyperpolarizing cell, bringing it back to resting potential
Hodgkin-Huxley Model

15. h switches off with increasing voltage
m is the fastest variable

17. The stochastic version may explain channel noise which affects the spiking timing reliability

19. K conductance is slow:
This allows the Na to raise the membrane potential before the K kicks in and hyper-polarizes the membrane

20. (Refinement: Goldman-Hodgkin-Katz current equations 2001)

21. Etc.

26. Electrical synapse:
Separation presynaptic and postsynaptic membrane is ~3.5nm
Joined by specific protein structures called gap junctions (specialized ionic channels that connect the cytoplasm of both cells)
Action potential comes to gap junction  depolarizes or hyperpolarizes the membrane  induces opening of the channels  diffusion of ions from one neuron to the other
Chemical synapse:
Separation is 20-40nm: no physical contact
Transmission is mediated by the release in this inter-synaptic space of neurotransmitters (generated at the end of the axon of the presynaptic neuron)
Action potential arrives at synapse  opening of some neurotransmitter vesicles near the membrane  release and diffusion of a large number of neurotransmitters  neurotransmitters have affinity for certain molecular receptors in the postsynaptic membrane  binding induces the opening of specific ionic channels  depolarization or hyperpolarizes of postsynaptic membrane + released vesicles replaced by others from a reserve pool of vesicles

27. Both cases result in a flux of ions through the postsynaptic membrane  synaptic current Isyn