Download Lecture 9

Neural Computation
Biological Neural Networks
Sources
• Material in this lecture comes from, “Handbook of
Natural Computing,” Editors Grzegorz Rosenberg,
Thomas Back and Joost N. Kok, Springer 2014.
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?
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)
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:
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
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
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
Open-probability of a channel depends on the voltage across the membrane
Action potential initiates:
1. Close to the threshold voltage a few Na channels start to open
2. [Na] higher outside the cell (because its reversal potential is +40mV) 
inflow Na, depolarizing cell
3. Opens even more Na channels and the spike initiates
4. Rapidly after spike starts, Na channels close again, K channels open
5. Outflow K, hyperpolarizing cell, bringing it back to resting potential
Hodgkin-Huxley Model
h switches off with increasing voltage
m is the fastest variable
The stochastic version may explain channel noise which affects the spiking timing reliability
K conductance is slow:
This allows the Na to raise the
membrane potential before the K
kicks in and hyper-polarizes the
membrane
(Refinement: Goldman-Hodgkin-Katz current equations 2001)
Etc.
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
Both cases result in a flux of ions through the postsynaptic membrane 
synaptic current Isyn