A brief introduction to
neuronal dynamics
Gemma Huguet
Universitat Politècnica de Catalunya
In Collaboration with David Terman
Mathematical Bioscience Institute
Ohio State University
Outline
Goal 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
Hodgin-Huxley model (1952)
Describe the generation of
action potentials in the
squid giant axon
Nobel Prize, 1963
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)
Na +
K+
K+
K+
Na +
Na +
Electrical signal
Open channel
Direction of propagation of nervous impulse
Closed channel
K+
K+
Cell
body
Resting and temporarily
unable to fire
Active state
(action potential)
Resting
Repolarization (K+)
0 mV
Travelling wave
-60 mV
Action potential
Action potential that propagates along
the axon
x
V
0 mV
-60 mV
Electrical activity of cells
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
C∂V/∂t = D ∂2V/∂x2 - Iion + Iapp
= D ∂2V/∂x2 - Σi gi (V-Vi)+Iapp
Hodgin-Huxley model (1952)
Model for electronically compact neurons V(x,t)=V(t).
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)
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 synapsis
Model for two mutually coupled neurons
dv1/dt = f(v1,w1) – gsyns2(v1 – vsyn)
Cell 1
dw1/dt = eg(v1,w1)
ds1/dt= a(1-s1)H(v1-q)-bs1
dv2/dt = f(v2,w2) – gsyns1(v2 – vsyn)
dw2/dt = eg(v2,w2)
ds2/dt = a(1-s2)H(v2-q)-bs2
 Assume si= H(vj-q), H Heaviside function
 (vi – vsyn) <0 (>0) excitatory (inhibitory) synapsis
Cell 2
Reduced model for chemical synapsis
Model for two mutually coupled neurons
dv1/dt = f(v1,w1) – gsyns2(v1 – vsyn)
Cell 1
dw1/dt = eg(v1,w1)
dv2/dt = f(v2,w2) – gsyns1(v2 – vsyn)
dw2/dt = eg(v2,w2)
Cell 2
s1= H(v1-q), s2 = H(v2-q)
 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.