A brief introduction to neuronal dynamics

Gemma Huguet

Universitat Politècnica de Catalunya

In Collaboration with David Terman

Mathematical Bioscience Institute

Ohio State University

OutlineGoal of mathematical neuroscience: develop and analyze

models for neuronal activity patterns.

1. Some biology

2. Modeling neuronal activity patterns

Single neuron models. Hodgkin-Huxley formalism.

Coupling between neurons. Chemical synapsis.

Network architecture.

3. Example. Numerical simulations of network activity patterns. Synchronization.

4. Conclusions.

The brain

~ 1012 Neurons

~ 1015 Synapses

How do we model neuronal systems?

The neuron

Electrical signal: Action potential that propagates along axon

Nobel Prize, 1963

Hodgin-Huxley model (1952)

Describe the generation of

action potentials in the

squid giant axon

Membrane potential The membrane cell separates two ionic solutions with different concentrations (ions are electrically charged atoms).

Membrane potential due to charge separation across the cell membrane.

V=Vin-Vout (by convention Vout=0)

Resting state V=-60 to -70 mV

Ionic channels embedded in the cell membrane (Na+ and K+ channels)

K +K +

K +

Na+

Na+ Na+

Closed channelOpen channelDirection of propagation of nervous impulse

RestingActive state(action potential)

Resting and temporarily unable to fire

Repolarization (K+)

K+

K+Cellbody

Electrical signal

Travelling wave

Action potential

0 mV

-60 mV

Action potential that propagates along the axon

xV

-60 mV

0 mV

Electrical parameters: • Potential Difference V(x,t)=Vin -Vout

• Current I(t)• Conductance g(t), Resistance R(t)=1/g(t) • Capacitance C

Rules for electrical circuits• Capacitor (Two conducting plates separated by an insulating layer. It stores charge). C dV/dt = I • Ohm´s Law I=Vg, IR=V

Current balance equation for membrane

Electrical activity of cells

C∂V/∂t = D ∂2V/∂x2 - Iion + Iapp

= D ∂2V/∂x2 - Σi gi (V-Vi)+Iapp

CdV/dt = - INa - IK – IL + Iapp

= – gNam3h(V-VNa) - gKn4(V-VK) - gL(V-VL) + Iapp

dm/dt = [m∞(V)-m]/m(V)dh/dt = [h∞(V) - h]/h(V)dn/dt = [n∞(V) – n]/n(V)

Hodgin-Huxley model (1952)

Model for electronically compact neurons V(x,t)=V(t).

V membrane potential

h,m,n channel state variables

Other models…

The models for single neurons are based on HH formalism.

Models for describing some activity patterns: silent, bursting, spiking.

Reduced models to study networks consisting of a large number of coupled neurons.

C dv/dt = f(v,w) + I dw/dt = εg(v,n)

Chemical synapsis

Synapsis can be:

Excitatory

Inhibitory

Presynaptic neuron

Postsynaptic neuron

Reduced model for chemical synapsisModel for two mutually coupled neurons

Assume si= H(vj-), H Heaviside function

(vi – vsyn) <0 (>0) excitatory (inhibitory) synapsis

dv1/dt = f(v1,w1) – gsyns2(v1 – vsyn)

dw1/dt = g(v1,w1)

ds1/dt= (1-s1)H(v1-)-s1

dv2/dt = f(v2,w2) – gsyns1(v2 – vsyn)

dw2/dt = g(v2,w2)

ds2/dt = (1-s2)H(v2-)-s2

Cell 1

Cell 2

Reduced model for chemical synapsisModel for two mutually coupled neurons

dv1/dt = f(v1,w1) – gsyns2(v1 – vsyn)

dw1/dt = g(v1,w1)

dv2/dt = f(v2,w2) – gsyns1(v2 – vsyn)

dw2/dt = g(v2,w2)

s1= H(v1-), s2 = H(v2-)

Cell 1

Cell 2

H Heaviside function ( H(x)=1 if x>0 and H(x)=0 if x<0 )

(vi – vsyn) <0 (>0) excitatory (inhibitory) synapsis

Network Architecture

Which neurons communicate with each other.

How are the synapsis: excitatory or inhibitory.

Exemple. Architecture of the STN/GPe network (Basal Ganglia, involved in the control of movement )

GPe CELLS

STN CELLS

Modeling neuronal activity patterns

Neuronal networks contain many parameters and time-scales:

• Intrinsic properties of individual neurons: Ionic channels.

• Synaptic properties: Excitatory/Inhibitory; Fast/Slow.

• Architecture of coupling.

Network activity patterns:

• Syncrhronized oscillations (all cell fires at the same time).

• Clustering (the population of cells breaks up into subpopulations; within each single block population fires synchronously and different blocks are desynchronized from each other).

• More complicated rythms

QUESTION: How do these properties interact to produce network behavior?

Example. Numerical simulations of network activity.

Clustering and propagating activity patterns

Synchronization

Why is synchronization important?

How do the brain know which neurons are firing according to the same reason?

Some diseases like Parkinson are associated to synchronization.

Conclusions

Goal of neuroscience: unsderstand how the nervous system communicates and processes information.

Goal of mathematical neuroscience: Develop and analyze mathematical models for neuronal activity patterns.

Mathematical models • Help to understand how AP are generated and how they can change as parameters are modulated.

• Interpret data, test hypothesis and suggest new experiments.

• The model has to be chosen at an appropriate level: complex to include the relevant processes and “easy” to analyze.