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.
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
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)
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?
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.