modeling small neural networks

Modeling Small Neural Modeling Small Neural NetworksNetworksBaktash BabadiBaktash [email protected]@ipm.ir

SCS, IPMSCS, IPMFall 2004Fall 2004

ReferencesReferences

Cokh & Segev (1998) Principles of neural Cokh & Segev (1998) Principles of neural modeling, 1998, MIT pressmodeling, 1998, MIT press

Abeles (1991) CorticonicsAbeles (1991) Corticonics

Ermentrout (1998) Neural Networks as Ermentrout (1998) Neural Networks as pattern forming Systems, Rep. Prog. Phys.pattern forming Systems, Rep. Prog. Phys.

……

Studying the Small Neural Studying the Small Neural Networks (1)Networks (1)

The dynamics of single neurons is not The dynamics of single neurons is not taken to be importanttaken to be important

The strength of synaptic connections is the The strength of synaptic connections is the important parameters important parameters

The dynamical state of the networks is the The dynamical state of the networks is the focus of attentionfocus of attention

Studying the Small Neural Studying the Small Neural Networks (2)Networks (2)

Network Architecture:Network Architecture:

Vi

Vj

Wi j

Ji j

Single Neuron network (1)Single Neuron network (1)

The neuron model:The neuron model:

Architecture:Architecture:

x

jjjii

i

exg

VWgVdt

dV

1

1)(

) ( ,

W


weightSynaptic:

Threshold:

W

xexg

VWgVdt

dV

1

1)(

).( The model:The model:

Parameters:Parameters:

Steady States:Steady States:).( ** VWgV


Steady States:Steady States:

).( ** VWgVVy

).( VWgy

WW

Bistability :Bistability :


Two Neuron Networks (1)Two Neuron Networks (1)

Architecture :Architecture :

Equations :Equations :

V1 V2W11 W22

W12

W21

xexg

VWVWgVdt

dV

VWVWgVdt

dV

1

1)(

) (

) (

222,212,122

2

121,211,111

1


Nullclines:Nullclines:

Shape of Nullclines:Shape of Nullclines:

Sigmoid:Sigmoid:

CubicCubic

xexg

VWVWgVdt

dV

VWVWgVdt

dV

1

1)(

) (

) (

222,212,122

2

121,211,111

1

) (

) (

222,212,12

121,211,11

VWVWgV

VWVWgV

Analyzed by Beer (1995), Ermentrout (1998):Analyzed by Beer (1995), Ermentrout (1998): Minimum number of fixed points: 1Minimum number of fixed points: 1

Maximum number of fixed points: 9Maximum number of fixed points: 9

The cubic nullclines have three rims The cubic nullclines have three rims (outer/inner/outer):(outer/inner/outer):

Intersections (fix points) :Intersections (fix points) :Outer-outer : StableOuter-outer : Stable

Inner-Inner: UnstableInner-Inner: Unstable

Inner-outer: Saddle pointInner-outer: Saddle point In general 13 dynamical states are possible in the two In general 13 dynamical states are possible in the two

neuron network.neuron network.


Example 1: Two stable fix points, one saddle Example 1: Two stable fix points, one saddle point: point:


Example: 2 saddle points, 3 stable fix Example: 2 saddle points, 3 stable fix points:points:


Example: 4 stable fix points, 4 unstable fix Example: 4 stable fix points, 4 unstable fix points, 1 saddle point:points, 1 saddle point:


Example: BistabilityExample: Bistability


Three Neuron Network (1)Three Neuron Network (1)

Architecture:Architecture:

Equations :Equations :

V1 V2

V3

xexg

VWVWVWgVdt

dV

VWVWVWgVdt

dV

VWVWVWgVdt

dV

1

1)(

) (

) (

) (

333,323,213,133

3

232,322,212,122

2

131,321,211,111

1

Chaotic Behavior is possible: Chaotic Behavior is possible:

Three Neuron Network (2)Three Neuron Network (2)

Netlets (1)Netlets (1)

In Netlets (Anninos et al 1970) The number In Netlets (Anninos et al 1970) The number of neurons is high, the number of of neurons is high, the number of connections is low.connections is low. Neuron Model:Neuron Model:

Architecture :Architecture :The number of neurons The number of neurons (N) (N) is highis high

Each neuron receives input from a Each neuron receives input from a

small number of other neurons (small number of other neurons (n=10n=10))

i

ii

jj

jii

h

ha

aWh

if 1

if 0

,


Dynamics:Dynamics: All the synaptic weights are equal All the synaptic weights are equal (W=1).(W=1). The time is taken to be discrete. We assume The time is taken to be discrete. We assume

that we observe the system at time steps that that we observe the system at time steps that are equal to the synaptic delay.are equal to the synaptic delay.

If a neuron fires at time If a neuron fires at time kk, it will be in , it will be in refractory period at time refractory period at time k+1.k+1.

Since the number of neurons is high, Since the number of neurons is high, statistical methods should be usedstatistical methods should be used


kk at time fired that neurons theoffraction The:?1 k

)1( , ,2

)(

: large and smallFor

2

2

2

)(

kkkk

m

nne

mp

n

kk mnmmnm

nmP

km

1)( )!(!

!)(

: at time inputs receivesneuron ay that Probabilit


th

dmmp

thm

)(

step last time the

from inputs receivesit y that probabilit The

1k at time firesneuron ay that Probabilit The

)1(

2

1

1

2)1(

)()1().1(

:1 at time fire that neurons offraction The

2

kk

k

n

nth

x

kk

th

kkk

dxe

dmmp

k

k

k

1

:1 at time firingfor neurons available offraction The


For For N=10N=10, , W=1W=1, th=5:, th=5:

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time


For For N=10N=10, , W=1W=1, th=3:, th=3:

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time


For For N=10N=10, , W=1W=1, th=1:, th=1:

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Netlets (8)Netlets (8)The iterative map :The iterative map :

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α k+1

α k

Th=3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Th=1α k+1

α k

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Th=5α k+1

α k

QuestionQuestion

How does the dynamics of a Netlet How does the dynamics of a Netlet change if not all the neurons that fired at change if not all the neurons that fired at time time kk, but a fraction of them (c) remain in , but a fraction of them (c) remain in refractory period at time refractory period at time k+1k+1??

Motor Pattern GeneratorsMotor Pattern Generators

The first article in Computational The first article in Computational Neuroscience:Neuroscience: Brown TG:Brown TG: On the nature of the fundamental activity On the nature of the fundamental activity

of the nervous centres; together with an analysis of of the nervous centres; together with an analysis of the conditioning of rhythmic activity in progression, the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the and a theory of the evolution of function in the nervous system.nervous system. J Physiol J Physiol 1914,1914,48:48:18-46.18-46.

The The

Reflex Loops vs. Central rhythm Reflex Loops vs. Central rhythm generationgeneration

Is the rhythmic motor patterns due toIs the rhythmic motor patterns due to reflex loops reflex loops Or a center that generates rhythms Or a center that generates rhythms

spontaneously? spontaneously?

Central Pattern Generators (CPG)Central Pattern Generators (CPG)

Central pattern generators (CPGs) are neural networks that can endogenously (i.e. without rhythmic sensory or central input) produce rhythmic patterned outputs; these networks underlie the production of most rhythmic motor patterns (Marder and Calabrese, 1996; Stein et al., 1997). The first modern evidence that rhythmic motor patterns are centrally generated was the demonstration that the locust nervous system, when isolated from the animal, could produce rhythmic output resembling that observed during flight.

Fictive MovementFictive Movement

A fictive motor pattern is a pattern of motor neuron firing that would, if the motor neurons were still attached to their muscles, result in the motor pattern in question being produced.

Mechanisms of Rhythm GenerationMechanisms of Rhythm Generation

1) Rhythms driven by Pace Maker 1) Rhythms driven by Pace Maker NeuronsNeurons Vertebrate Respiratory SystemVertebrate Respiratory System Pyloric Ganglion of CrustaceanPyloric Ganglion of Crustacean

2) Rhythms resulting from synaptic 2) Rhythms resulting from synaptic interactions of the neurons. interactions of the neurons. Usually emerge from mutually inhibitory Usually emerge from mutually inhibitory

neurons (reciprocal inhibition).neurons (reciprocal inhibition). Called “Half Center Oscillators” Called “Half Center Oscillators”

Half Center OscillatorsHalf Center Oscillators

The mechanisms of The mechanisms of transition between transition between activation and activation and inhibition:inhibition: Spike frequency Spike frequency

adaptationadaptation Escape from Escape from

inhibitioninhibition

Cellular Properties of Neurons in Cellular Properties of Neurons in CPGCPG

a) Spontaneous Rhythmic firinga) Spontaneous Rhythmic firing

b) Plateau Firingb) Plateau Firing

c) Escape from inhibitionc) Escape from inhibition

d) Post inhibitory reboundd) Post inhibitory rebound

e) Delayed Post inhibitory Rebounde) Delayed Post inhibitory Rebound

Example: Somatogastric (STG) Example: Somatogastric (STG) Nervous System of CrustaceansNervous System of Crustaceans

Marder & Abbott 1997, Modeling small Marder & Abbott 1997, Modeling small neural networks:neural networks:

SourceSource

Precise Firing Sequences (PFS)Precise Firing Sequences (PFS)

Prut et al, 1998:Prut et al, 1998:

Synfire ChainsSynfire Chains

The reproducibility of PFSs implies that The reproducibility of PFSs implies that there are synchronous pools of neurons in there are synchronous pools of neurons in the cortex (Abeles 1991).the cortex (Abeles 1991).

The Notion of Synfire ChainsThe Notion of Synfire Chains Based on anatomical and physiological Based on anatomical and physiological

data, Abeles(1991) proposed the data, Abeles(1991) proposed the Synfire Synfire model which is:model which is:

A locally feed-forward neural networkA locally feed-forward neural network With convergent/divergent connectionsWith convergent/divergent connections

pool

……

Link

……

Propagation of synchronous patterns in Synfire Chains is an explanation for precise firing sequences

Dynamics of Firing Patterns in Dynamics of Firing Patterns in Synfire Chains (1)Synfire Chains (1)

If the Synfire notion is true, the neural activity in successive pools must tend to synchronize and remain synchronous

Hermann, Hertz , Prugel-Bennett (1996): In a simple synfire

model of non-leaky integrate-and-fire neurons, the firing patterns in successive pools tend to synchronize.

Pulse Packets in Synfire networksPulse Packets in Synfire networks

In order to study the spike synchronization In order to study the spike synchronization in synfire networks, Aertsen et al (1995) in synfire networks, Aertsen et al (1995) introduced the notion of pulse packet:introduced the notion of pulse packet: A pulse packet is an index of the activity of a A pulse packet is an index of the activity of a

neuron pool and is defined by two neuron pool and is defined by two parameters:parameters:

1) The number of neurons that fire in the pool1) The number of neurons that fire in the pool

2) The standard deviation of the firing times2) The standard deviation of the firing times

Dynamics of Firing Patterns in Dynamics of Firing Patterns in Feed-forward Networks (1)Feed-forward Networks (1)

Diesmann, Gewaltig, Aertsen (1999)Diesmann, Gewaltig, Aertsen (1999)

Bistability in a phase-plane portrait: Dense and highly synchronous pulse packets

will propagate successfully. Sparse and weakly synchronous ones will

eventually dissipate.

Gewaltig, Diesman, Aertsen (2001)Gewaltig, Diesman, Aertsen (2001)

Survival probability of pulse packets in single trials:

Dynamics of Firing Patterns in Dynamics of Firing Patterns in Feed-forward Networks (2)Feed-forward Networks (2)

Dynamics of Firing Patterns in Dynamics of Firing Patterns in Synfire Chains (3)Synfire Chains (3)

Cateau & Fukai (2001) : Using Fokker-Planck equations, the

previous results have been confirmed analytically.

Common Issues Among All the Common Issues Among All the Mentioned StudiesMentioned Studies::

The Synfire activity ends in two scenarios:

Saturation : All the neurons in the final pools fire

Decay: No neuron in the final pools fire (apart from the background activity)

In either case there is a loss of information content

The synaptic weights are uniform

Our QuestionOur Question

Is it possible to avoid full saturation and Is it possible to avoid full saturation and full decay by modifying the synaptic full decay by modifying the synaptic weights of a Synfire network?weights of a Synfire network?

Analysis Assumptions :Analysis Assumptions :

While we do not aim to study the While we do not aim to study the synchronizing property of the network and synchronizing property of the network and only the number of firing neurons is our only the number of firing neurons is our mater of interest, the network is fed with a mater of interest, the network is fed with a fully synchronous pattern.fully synchronous pattern.

No synaptic delay is taken into account.No synaptic delay is taken into account.

Reformulating the problem:Reformulating the problem:

Assume that we feed the net with an input pattern

containing n firing neurons. How should we set the weight parameters ( , ) to avoid full saturation (final n=50) or full decay (final n=0)?

w w

Method of AnalysisMethod of Analysis

Iterative mappingIterative mapping )( 1 nRn

The Iterative MapThe Iterative Map

)( )( 1

0

nRduuGNn

u

2221

1

0 thwn

wnthu

2)(

2

2u

euG

When Mean Weight is positive:

Three attractor states are possible for the number of neurons:

Full saturation / full decay behavior

Only full fade behavior

Attraction to a fixed number of firing neurons

Full saturation and full decayFull saturation and full decay

When :When : 0.001 0.003 ww

Only full decayOnly full decay

When :When : 0.001 0.015 ww

Fixed point AttractorFixed point Attractor

When :When : 0.02 0.003 ww

Three attractor states are possible for Three attractor states are possible for the number of neurons:the number of neurons:

Attraction to a fixed number of firing Attraction to a fixed number of firing neuronsneurons

Oscillation between different number of Oscillation between different number of firing neuronsfiring neurons

Multiple periodic/chaotic behaviorMultiple periodic/chaotic behavior

When Mean Weight is Negative:When Mean Weight is Negative:

Fixed point attractorFixed point attractor

When :When : 0.64 0.3- ww

Cycle AttractorCycle Attractor

When :When : 0.528 3.0 ww

Multiple periodic/Chaotic attractorMultiple periodic/Chaotic attractor When :When :

0.256 3.0 ww

The Bifurcation Diagram for The Bifurcation Diagram for Negative Mean WeightNegative Mean Weight

3D bifurcation diagram ! 3D bifurcation diagram !

3D bifurcation diagram !3D bifurcation diagram !

Two further Questions:Two further Questions:

What if we feed the network with What if we feed the network with asynchronous inputs?asynchronous inputs?

What if the neurons have spontaneous What if the neurons have spontaneous firings?firings?

The Simulation results with The Simulation results with asynchronous inputsasynchronous inputs

The Simulation results with The Simulation results with asynchronous inputs and asynchronous inputs and

spontaneous firingsspontaneous firings

Open Questions:Open Questions:

1) The speed of pulse propagation in 1) The speed of pulse propagation in synfire chainssynfire chains

2) Analysis of asynchronous pulse packets 2) Analysis of asynchronous pulse packets in synfire chainsin synfire chains

3) The effect of synaptic plasticity on the 3) The effect of synaptic plasticity on the dynamics of synfire chainsdynamics of synfire chains

modeling small neural networks

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