Transcript Computational neuroscience - School of Computing and Mathematics

Peter Andras
School of Computing and Mathematics, Keele University
COMPUTATIONAL NEUROSCIENCE
OVERVIEW
The brain
 Neurons and information
 Computational models
 Mathematical and computational analysis
 Back to biology

2
BRAIN FUNCTION
The nervous system
controls the behaviour of
animals
 The brain is a collection
of high level specialised
neural centres (ganglia)

3
BRAIN FUNCTION



Sensory brain: interpreting
visual, auditory, somatosensory, olfactory, etc
information
Motor brain: high level control
of muscles – large and fine
scale control
Association brain: linking
sensory and motor function,
managing memories, making
general sense of the world,
driving communications
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COGNITIVE BRAIN

Understanding the world
 Perception
 Action
 Decision
making
 Memories
 Learning

Black box models
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BRAIN DISEASE

Parkinson’s Disease

Alzheimer’s Disease

Creutzfeldt – Jakob Disease
(mad cow disease)
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BRAIN STRUCTURE
Large-scale connectivity
– networks of brain
modules
 Layers of neurons

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NEURONS
Neurons are the building
blocks of the nervous
system
 Synapses mediate
communication between
neurons
 Synapses may form, get
strengthened or weakened,
or may disappear

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NEURAL ACTIVITY





Neural cell membrane differential
permeability for ions – Na+, K+, Cl-, Ca++
Ionic imbalance leads to steady state potential
difference: ~-70 mV the inside is more
negative than the outside
Neurotransmitters trigger the opening of ionic
channels, also voltage-dependent channels,
electric junctions (fully or partly bi-directional)
The membrane potential changes and this
propagates along the membrane  dendritic
signals, action potentials (spikes) in the axons
Spiking activity can be triggered by several
mechanisms – e.g. excitatory input, rebound
from inhibition
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NEURONS AND INFORMATION
Information may be encoded in the rate of spiking
– e.g. sensory neurons, motor neurons innervating
muscles
 Information may be encoded in the temporal
pattern of spikes – e.g. some projection neurons in
the cortex
 Information may be represented by spatiotemporal patterns of activity of many neurons –
e.g. olfactory bulb, hippocampus – short term
memory formation

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NEURAL CIRCUITS AND NETWORKS

Neurons are organised in
functional blocks

A neuron may belong to
multiple functional blocks

Hierarchical combination
of functional blocks
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NEUROMODULATION
E.g. Dopamine, Serotonin, Noradrenalin, Oxytocin
 Generally neuromodulators alter directly or
indirectly the functioning of ion channels
modulating the behaviour of neurons
 Neuromodulators may also have long-term effects
by influencing the transcription of the DNA
 Neuromodulators determine the active parts of
anatomical networks  many functional networks
may be supported by the same anatomical
network under different neuromodulation

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MODEL ANIMAL SYSTEMS




C. Elegans – network
organisation, development,
sensory – motor coordination
Crab / lobster stomatogastric
ganglion – neuromodulation,
motor control – central pattern
generator, autonomous
functional restoration
Aplysia – memory and learning
Drosophila – complex
behaviour, development
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COMPUTATIONAL MODELS

Simple models – perceptron: 0 / 1 – active /
inactive
Classification
theory

Networked models – nonlinear, multi-layer
perceptrons
Nonlinear
approximation
theory
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COMPUTATIONAL MODELS

More realistic models based on ionic current
conductances and modelling of ionic currents
 Hodgkin
– Huxley (HH) model
dV
C
I
dt
I  I 0  I Na  I K  I Ca  I Cl
I ion  g ion  m a  h b  (V  Eion )
dm m (V )  m

dt
 m (V )
dh h (V )  h

dt
 h (V )
g ion  ion
specific
maximal conductanc e
m  probabilit y of open activation gate
h  probabilit y of open inactivati on gate
a  number of activation gates
b  number of inactivati on gates
Eion  ion specific equilibriu m potential
m , h  steady state activation / inactivati on functions,
voltage dependent
 m , h  voltage dependent time constants
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COMPUTATIONAL MODELS

Original Hodgkin – Huxley model
dV
C
 I 0  g Na  m3  h  (V  E Na )  g K  n 4  (V  EK )  g L  (V  EL )
dt
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COMPUTATIONAL MODELS

dV
 a V 3  b V 2  y  z  I
dt
dy
 c  d V 2 y
dt
dz
 r  ( s  (V  E )  z ); r  0.001
dt
Simplified models
 Hindmarsh
 FitzHugh
– Rose
dV
 V  (a  V )  (V  1)  w  I
dt
dw
 b V  c  w
dt
– Nagumo
dV
 I  g L  (V  EL )  g Ca  m  (V  ECa )  g K  n  (V  EK )
dt
dn n  n

dt
n
C
 Morris
– Lecar
m 
1
2e
V V1
 2
V2
1
, n 
2e
V V3
 2
V4
, n 
1
 V  V3 

  cosh 
2
V
4 

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MODEL ANALYSIS
dx
0
dt

Variable – corresponding nullclines –

Intersections of nullclines  nodes, saddles, focuses, saddle-nodes


Limit cycles – periodic trajectories


Stable equilibrium points imply convergence to a steady state
Activity along a limit cycle may correspond to sub-threshold oscillations
or spiking behaviour
Unstable
Focus
Phase plane analysis
nullcline
Stable
node
nullcline
Saddle
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MODEL ANALYSIS



Depending on external input (I 0 ) the nullclines shift and
the system that converged previously to a stable node or
focus experiences a change moving it onto a limit cycle
trajectory  silent neuron becomes a spiking neurons
Alternatively the system may be on small scale limit
cycle and switches to a larger size limit cycle  neuron
with sub-threshold activity starts spiking
Reverse transition: the spiking stops
Stable
node
Saddle
Saddle-node
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MODEL ANALYSIS

Bifurcation analysis – how is the qualitative
behaviour of the system changing as the
parameters change (e.g. external input current)
Two heteroclinic orbits
One periodic
homoclinic orbit
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NUMERICAL ANALYSIS

Combined slow and fast dynamics – requires
adaptive integration step choice

Sensitivity to numerical precision of
calculations

Numerical problems grow when simulated
neurons get coupled into simulated neural
circuits
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NUMERICAL ANALYSIS

Many parameter combinations correspond to the same
behaviour in the modelled neuron



Exhaustive search of the parameter space – problem many
parameters imply high dimensional parameter space,
exponential growth of required samples
Experimental data shows correlations between parameters
– use these to reduce the dimensionality and size of the
parameter space
Different parameter combinations may produce the same
basic behaviour but do not produce realistic behaviour in
other circumstances (e.g. exposure to neurotoxins or
neuromodulators, integration into a model neural circuit)
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EXAMPLE 1

Is nonlinearity in inward current required
for spiking model neurons? (Bose, A
Golowasch, J, Guan, Y, Nadim, F (2014) J
Comput Neurosci, 37:229-242)
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EXAMPLE 1

Is nonlinearity in inward current required
for spiking model neurons? (Bose, A ,
Golowasch, J, Guan, Y, Nadim, F (2014) J
Comput Neurosci, 37:229-242)
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CENTRAL PATTERN GENERATORS

Motor control
 Movements
of muscles are composed from
rhythmic movements

Rhythmic movements are generated by neural
circuits called central pattern generators
 E.g.
respiration, mastication, swallowing
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CENTRAL PATTERN GENERATORS

Model:
 Pacemaker
neuron: autonomous rhythm generator
 Reciprocally inhibiting neurons – half centre
oscillator

Half-centre oscillator
 Escape:
the inhibited neuron’s behaviour changes
and escapes from inhibition
 Release: the inhibiting neuron’s behaviour changes
and the other neuron gets released from the
inhibition
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EXAMPLE 2

Analysis of CPGs with half-centre oscillators (Daun, S. Rubin, JE,
Rybak, IA (2009), J Comput Neurosci, 27: 3-26)

Escape:
dV
  g Na  m (V )  h  (V  E Na )  g L  (V  EL )  g syn  s  (V  Esyn )  g a V
dt
dh h (V )  h

dt
 h (V )
C
ds
   (1  s )  s (V )    s
dt
m (V ) 
1
v  m
1 e
1
, h (V ) 
m
v  h
1 e
h
 2(v   h ) 

,  h    cosh 

h


1
, s (V ) 
v  syn
1 e
 syn
dV
  gT  m (V )  h  (V  ECa )  g L  (V  EL )  g syn  s  (V  Esyn )  g a V
dt
dh h (V )  h

dt
 h (V )
C

Release:
ds
   (1  s )  s (V )    s
dt
1
m (V ) 
, h (V ) 
v  m
1 e

m
1
1 e

v  h
h
1
, s (V ) 
1 e

v  syn
 syn
,  h  t0 
t1
1 e

v  ht
 ht
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EXAMPLE 2

Analysis of CPGs with half-centre oscillators (Daun, S. Rubin, JE,
Rybak, IA (2009), J Comput Neurosci, 27: 3-26)

Escape

Release
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OVERLAPPING NEURAL CIRCUITS

There is indirect evidence that neurons belong
to multiple functional circuits in many parts of
the nervous systems
 E.g.
place cells, grid cells, neurons in the primary
visual cortex, swimming neurons in marine snails

What are the mechanisms of such neuronal
behaviour ?
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EXAMPLE 3

Modelling and analysis of functional
switching of neurons between rhythm
generating circuits (Gutierrez, GJ, O’Leary,
T, Marder, E (2013), Neuron, 77: 845858.)
Crustacean STG with
pyloric and gastric
rhythm networks and
the IC neuron at the
intersection of these
networks
Model network with a hub
neuron that may belong
functionally to the two halfcentre oscillator sub-networks
(red and blue / fast and slow
half-centre oscillators)
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EXAMPLE 3

Modelling and analysis of functional
switching of neurons between rhythm
generating circuits (Gutierrez, GJ, O’Leary,
T, Marder, E (2013), Neuron, 77: 845858.)
dV
  g L  (V  EL )  g Ca  m (V )  (V  ECa )  g K  n  (V  EK )  g h  h  (V  Eh )  g el  (V  V other )  g syn  s (V )  (V  Vsyn )
dt
dn
dh h (V )  h
 n (V )  (n (V )  n),

dt
dt
 h (V )
C
m (V ) 
h (V ) 
 V  V1 
 V  V3 
 V  V3 
1
1
), n (V )   (1  tanh 
), n (V )  n  cosh 

 (1  tanh 
2
V
2
V
2
V
4 
 2 
 4 

1
V V5
V6
,  h (V )  272 
1499
V V7
V8
, s (V ) 
1
V V9
V10
1 e
1 e
1 e
V1  0, V2  20,V3  0,V4  15,V5  78.3, V6  10.5,V7  42.2,V8  87.3, V9  25, V10  5
n  0.002, g el  [0.25,7.5], g syn  [0.25,10], g L  0.0001
gCa
Fast neurons
 0.019, g K  0.039, g h  0.025
gCa
gCa
Slow neurons
 0.0085, g K  0.015, g h  0.01
Hub neuron
 0.017, g K  0.019, g h  0.008
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EXAMPLE 3

Modelling and analysis of
functional switching of
neurons between rhythm
generating circuits
(Gutierrez, GJ, O’Leary, T,
Marder, E (2013), Neuron,
77: 845-858.)
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MODELLING NEUROMODULATION

It has been shown that neuromodulators can have a
global impact on a network that is different from the
sum of their impact on individual sepate neurons (e.g.
Hooper and Marder, 1987, J Neurosci, 7: 2097-2112)
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MODELLING NEUROMODULATION



Inclusion of modulator induced ionic currents into neuron models
Difficult to assess network effect
New data: simultaneous VSD recording of many identified
neurons exposed to neuromodulation
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VARIABILITY AND ROBUSTNESS
Many parameter settings deliver the same
model neuron behaviour
 Parameter correlations are determined
experimentally
 Relatively small changes of parameters by
neuromodulators may induce significant
behavioural changes in individual neurons or
the network of neurons

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VARIABILITY AND ROBUSTNESS
Investigation of the role of parameter variability
in reproducing realistic network behaviour
 Reproduction of the impact of neuromodulators
and the analysis of changing roles of identified
neurons in the context of the network

-30
-35
-40
-45
-50
-55
-60
4000
4100
4200
4300
4400
4500
4600
4700
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PHASE LOCKING

Synchronisation of weakly coupled oscillators
 Oscillator
= dynamical system moving along a limit
cycle attractor
 Coupling = synaptic and electrical connections
N
d i
 i   aij  ( j   i )
dt
j 1
 i  phases, i  frequencies
Generally: phase locking – can be the same or
opposite phase or other phase relationship
 Neuromodulation of phase locking

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LEADING TO BIOLOGICAL HYPOTHESES

Predictions about
 the
roles and nature of ionic currents in neurons
 the joint roles of neurons in the context of neural
circuits
 the mechanisms underlying the individual and joint
roles of neurons
 possible interpretations of experimental data
38
LEADING TO BIOLOGICAL HYPOTHESES

Examples:
multiple parameter values lead to similar neural
behaviour  experimental testing led to the realisation
of correlations between parameters
 computational models of grid cells suggested a
universal kind of position encoding by grid cells in the
entorhinal cortex, which recently has been checked and
rejected
 computational models of neurons predicted behaviours
of networks that were not confirmed experimentally 
highlighting the role of neuromodulators and directing
experimental investigations toward the study of impact
of neuromodulators on network level behaviour

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LEADING TO BIOLOGICAL HYPOTHESES



Often the predictions based on computational models
are wrong, i.e. not confirmed or supported by the
biological data
However such wrong predictions underline the
conceptual errors in the biological and functional
understanding of neural systems and direct the
experimental work in directions that can provide
elucidating answers and ultimately corrections of the
previous wrong assumptions
Some predictions based on mathematical and
computational analysis of course turn out to be correct
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CONCLUSIONS




Biological neural systems are very complex and difficult to
understand
Computational modelling and mathematical analysis of
models of neurons and neural circuits helps the
understanding of how biological neural systems work
Bio-realistic modelling of neurons using conductance-based
models are useful in particular both in terms of readiness for
mathematical and computational analysis and in terms of
biological relevance and ease of biological interpretation
Often predictions based on computational models and
analysis are wrong, but even in such cases they contribute
very much for the direction of experimental research towards
questions that lead to much improved understanding of
biological neural systems
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ACKNOWLEDGEMENTS

Newcastle University
 Jannetta
Steyn (PhD student)
 Thomas Alderson (MSc student)

Illinois State University
 Dr
Wolfgang Stein (PI)
 Carola Staedele (PhD student)
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