A brief introduction to neuronal dynamics

1. A brief introduction to neuronal dynamics Gemma Huguet Universitat Politècnica de Catalunya In Collaboration with David Terman Mathematical Bioscience Institute Ohio State University

2. 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. • Example. Numerical simulations of network activity patterns. Synchronization. • Conclusions.

3. The brain ~ 1012 Neurons ~ 1015 Synapses How do we model neuronal systems?

4. The neuron Electrical signal: Action potential that propagates along axon

5. Hodgin-Huxley model (1952) Describe the generation of action potentials in the squid giant axon Nobel Prize, 1963

6. + + + Na Na Na + + + K K K 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)

7. Open channel Closed channel Direction of propagation of nervous impulse K+ K+ Cell body Active state (action potential) Resting Repolarization (K+) Resting and temporarily unable to fire Electrical signal 0 mV Travelling wave -60 mV Actionpotential

8. Action potential that propagates along the axon x V 0 mV -60 mV

9. 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

10. 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) Vmembrane potential h,m,nchannel state variables

11. 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)

12. Chemical synapsis • Synapsis can be: • Excitatory • Inhibitory Presynaptic neuron Postsynaptic neuron

13. Reduced model for chemical synapsis Model for two mutually coupled neurons dv1/dt = f(v1,w1) – gsyns2(v1 – vsyn) 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 Cell 1 Cell 2 • Assume si= H(vj-q), H Heaviside function • (vi – vsyn) <0 (>0) excitatory (inhibitory) synapsis

14. Reduced model for chemical synapsis Model for two mutually coupled neurons dv1/dt = f(v1,w1) – gsyns2(v1 – vsyn) dw1/dt = eg(v1,w1) dv2/dt = f(v2,w2) – gsyns1(v2 – vsyn) dw2/dt = eg(v2,w2) s1= H(v1-q), s2 = H(v2-q) 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

15. 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

16. 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?

17. Example. Numerical simulations of network activity. Clustering and propagating activity patterns

18. 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.

19. 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.