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

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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.

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The brain ~ 10 12 Neurons ~ 10 15 Synapses How do we model neuronal systems?

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The neuron Electrical signal: Action potential that propagates along axon

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

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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=V in -V out (by convention V out =0) Resting state V=-60 to -70 mV Ionic channels embedded in the cell membrane (Na + and K + channels) K + K + K + Na + + +

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

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

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Electrical parameters: Potential Difference V(x,t)=V in -V out 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 ∂ 2 V/∂x 2 - I ion + I app = D ∂ 2 V/∂x 2 - Σ i g i (V-V i )+I app

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CdV/dt = - I Na - I K – I L + I app = – g Na m 3 h(V-V Na ) - g K n 4 (V-V K ) - g L (V-V L ) + I app 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

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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)

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Chemical synapsis Synapsis can be: Excitatory Inhibitory Presynaptic neuron Postsynaptic neuron

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Reduced model for chemical synapsis Model for two mutually coupled neurons Assume s i = H(v j - ), H Heaviside function (v i – v syn ) 0) excitatory (inhibitory) synapsis dv 1 /dt = f(v 1,w 1 ) – g syn s 2 (v 1 – v syn ) dw 1 /dt = g(v 1,w 1 ) ds 1 /dt= (1-s 1 )H(v 1 - )- s 1 dv 2 /dt = f(v 2,w 2 ) – g syn s 1 (v 2 – v syn ) dw 2 /dt = g(v 2,w 2 ) ds 2 /dt = (1-s 2 )H(v 2 - )- s 2 Cell 1 Cell 2

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Reduced model for chemical synapsis Model for two mutually coupled neurons dv 1 /dt = f(v 1,w 1 ) – g syn s 2 (v 1 – v syn ) dw 1 /dt = g(v 1,w 1 ) dv 2 /dt = f(v 2,w 2 ) – g syn s 1 (v 2 – v syn ) dw 2 /dt = g(v 2,w 2 ) s 1 = H(v 1 - ), s 2 = H(v 2 - ) Cell 1 Cell 2 H Heaviside function ( H(x)=1 if x>0 and H(x)=0 if x<0 ) (v i – v syn ) 0) excitatory (inhibitory) synapsis

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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

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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?

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

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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.

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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.

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