Basics of Computational Neuroscience: Neurons and Synapses to Networks

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1 Basics of Computational Neuroscience: Neurons and Synapses to Networks Bruce Graham Mathematics School of Natural Sciences University of Stirling Scotland, U.K. Useful Book Authors: David Sterratt, Bruce Graham, Andrew Gillies, David Willshaw Cambridge University Press, 2011 Companion website at: compneuroprinciples.org 2 1

2 Whole brain Brain nuclei Lumped models Networks of neurons Single neurons Subcellular Levels of Detail (confrontaal.org) (Fig. 1.3 pg 7) 3 Neural Networks Networks of complex neurons Pulse train signals (action potentials or spikes) Dynamic connection weights Y 1 Y 2 Y k O 1 O j X 1 X 2 X 3 X i 4 2

3 Complicated Neural Circuits CA1 region of hippocampus (Klausberger & Somogyi, 2008) 5 Neurons Neurons come in many shapes and sizes (Dendrites, Hausser et al (eds)) 6 3

4 Why Model a Neuron? Response to inputs from other neurons? Membrane potential Intrinsic membrane properties Synaptic signal integration SLM INPUT SR OUTPUT SP SO 7 Compartmental Modelling Membrane potential (Fig. 4.1 pg 73) 8 4

5 Electrical Potential of a Neuron Differences in ionic concentrations Transport of ions Sodium (Na) Potassium (K) (Fig. 2.1 pg 14) (Fig pg 31) 9 A Model of Passive Membrane A resistor and a capacitor Kirchhoff s current law (Fig pg 32) 10 5

6 A Length of Membrane Membrane compartments connected by intracellular resistance Compartmental modelling equation (Fig pg 36) 11 The Action Potential Output signal of a neuron Rapid change in membrane potential Flow of Na and K ions (Fig. 3.1 pg 47) 12 6

7 Action Potential Model Empirical model by Hodgkin and Huxley, 1952 Voltage-dependent Na and K channels (Fig. 3.1 pg 47) (Fig. 3.2 pg 50) 13 Time Varying Conductances K conductance: n a function of time and voltage (Fig pg 63) 14 7

8 Complete Action Potential Model Box 3.5 pg 61 (Fig pg 60) 15 Propagating Action Potential (Fig pg 65) 16 8

9 Families of Ion Channels Sodium (Na): fast, persistent Potassium (K): delayed rectifier, A, M Calcium (Ca): low and high voltage activated L, N, R, T Calcium-activated potassium: sahp, mahp Non-specific cation: H Around 140 different voltage-gated ion channel types. A neuron may express 10 to 20 types. 17 Potassium A-current: K A Different characteristics from delayed rectifier: K DR Low threshold activating / inactivating current 18 9

10 One Effect of A-current Type I: with K A Steady increase in firing frequency with driving current Type II: without K A Suddent jump to non-zero firing rate (Fig. 5.9 pg 106) 19 Large Scale Neuron Model Pyramidal cell Hippocampal pyramidal cell (PC) 20 10

11 Detailed Pyramidal Cell Model 183 electrical compartments Heterogeneous ion channel population (Poirazzi & Pissadaki, in Hippocampal Microcircuits) 21 Pyramidal Cell Model Responses Reproduces somatic and dendritic current injection experimental results Sodium spiking with distance (Poirazzi & Pissadaki) 22 11

12 Varying Levels of Detail Capture essential features of morphology Simple structure apical distal proximal soma basal 23 Reduced Pyramidal Cell Model 2-compartment model Pinsky & Rinzel (1994) Captures essence of PC behaviour Single spikes and bursting Soma Area p g Area 1-p Dendrite (Fig. 8.1 pg 199) (Box 8.1 pg 200) 24 12

13 Pinsky-Rinzel Model in Action Behaviour depends on Compartment coupling strength (g) Magnitude of driving current (I) Low I, low g High I, low g High I, high g (Fig. 8.2 pg 201) 25 Simple Spiking Neuron Models Simplified equations for generating action potentials (APs) FitzHugh-Nagumo; Kepler; Morris-Lecar 2 state variables: voltage plus one other H-H model contains 4 variables: V, m, h, n Simple spiking models that DO NOT model the AP waveform Integrate-and-fire models 26 13

14 Integrate-and-Fire Model RC circuit with spiking and reset mechanisms When V reaches a threshold A spike (AP) event is signalled Switch closes and V is reset to E m Switch remains closed for refractory period (Fig. 8.4 pg 204) 27 I&F Model Response Response to constant current injection No refractory period 10ms refractory period (Fig. 8.5 pg 205) 28 14

15 More Realistic I&F Neurons Basic I&F model does not accurately capture the diversity of neuronal firing patterns Adaptation of interspike intervals (ISIs) over time Precise timing of AP initiation Noise (Fig. 8.8 pg 213) 29 Modelling AP Initiation Basic I&F is a poor model of the ionic currents near AP threshold Quadratic I&F Exponential I&F (Fig. 8.9 pg 214) 30 15

16 Quadratic I&F plus dynamic recovery variable The Izhikevich Model (Fig pg 215) 31 Neural Connections: Synapses (Fig. 7.1 pg 173) 32 16

17 Synaptic Conductance 3 commonly used simple waveforms a) Single exponential b) Alpha function c) Dual exponential Current: I syn (t) = g syn (t)(v(t)-e syn ) (Fig. 7.2 pg 174) 33 Neuronal Firing Patterns Neuronal firing activity is often irregular How does this arise? Intrinsic or network property? Balance of excitation and inhibition Excitation Inhibition Output 34 17

18 Model using I&F Neuron I&F neuron driven by 100Hz Poisson spike trains Via excitatory and inhibitory synapses Alter balance of excitation and inhibition 300 Excitation inhibition 18 Excitation only Irregular firing More regular (Fig. 8.6 pg 209) 35 Network Model with I&F Neurons Randomly connected network of 80% excitatory and 20% inhibitory neurons External excitatory drive to all neurons Noisy Poisson spike trains 36 18

19 Network Model: Random Firing 37 Rhythm Generation E-I oscillator Reciprocally coupled excitatory and inhibitory neurons Constant drive to excitatory neuron Delay around the loop 38 19

20 Learning in the Nervous System ANNs learn by adapting the connection weights Different learning rules Real chemical synapses do change their strength in response to neural activity Short-term changes Milliseconds to seconds Not classified as learning Long term potentiation (LTP) and depression (LTD) Changes that last for hours and possibly lifetime Evidence that LTP/LTD corresponds to learning 39 Hebbian Learning Hypothesis by Donald Hebb, The Organization of Behaviour, 1949 When an axon of cell A excites cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells so that A s efficiency, as one of the cells firing B, is increased. A B 40 20

21 Associative Learning: Hebbian Increase synaptic strength if both pre- and postsynaptic neurons are active: LTP Decrease synaptic strength when the pre- or postsynaptic neuron is active alone: LTD 41 Example: Associative Memory Autoassociation and heteroassociation Hebbian learning of weights Content addressable 42 21

22 Heteroassociative Memory Associations between binary patterns Hebbian learning: w ij = p i.p j Store multiple patterns (Fig. 9.4 pg 234) 43 Memory Recall Weighted synaptic input from memory cue Threshold setting on output (Fig. 9.5 pg 235) 44 22

23 Multistep Memory Recall Autoassociative recurrent network (Fig. 9.6 pg 236) 45 Spiking Associative Network How could this be implemented by spiking neurons? Sommers and Wennekers (2000, 2001) 100 Pyramidal cell recurrent network Pinsky-Rinzel 2-compartment PC model E connections determined by predefined binary Hebbian weight matrix that sets AMPA conductance All-to-all fixed weight inhibitory connections Tests autoassociative memory recall 46 23

24 Spiking Associative Network Pattern is 10 active neurons out of random patterns stored 4 active neurons as recall cue Excitatory connections via Hebbian learning All-to-all inhibitory connections 47 Cued Recall in Spiking Network Cue: 4 of 10 PCs in a stored pattern receive constant excitation Network fires with gamma frequency Pattern is active cells on each gamma cycle Timing and strength of inhibition CUE PATTERN SPURIOUS (Fig pg 253) 48 24

25 To Follow PRACTICAL WORK: Simulating neurons and neural networks with NEURON Plasticity in the nervous system Spike-time-dependent plasticity (STDP) Other scales (spatial and temporal) to model Other signals Extracellular field potentials 49 25