Dynamics of Networks 1Basic Formalism &

Symmetry

Ian Stewart

Mathematics Institute

University of Warwick

UK-Japan Winter SchoolDynamics and Complexity

Examples of Network Dynamics

nerve cell or neuron

Examples of Network DynamicsNeurons form networks that transmit and process signals

Examples of Network DynamicsIndividual neurons can be modelled by an ODE

Hodgkin-Huxley Equations

Fitzhugh-Nagumo Equations

Morris-Lecar Equations

and many others...

Examples of Network DynamicsFitzhugh-Nagumo Equations

dv/dt = v(a-v)(v-1) - w + Ia

dw/dt = bv - w

v = membrane potential

w = substitute for ion channel variables

Ia = applied current

a, b, are constants

0 < a < 1 b, ≥ 0

Examples of Network DynamicsFitzhugh-Nagumo Equations

dv/dt

= v(a-v)(v-1)

- w + Ia

dw/dt

= bv - w

Examples of Network DynamicsCoupled Fitzhugh-Nagumo Equations for 2 neurons

dv1/dt = v1 (a-v1)(v1-1) - w1

dw1/dt = bv1 - w1

Ia = 0

dv2/dt = v2 (a-v2)(v2-1) - w2

dw2/dt = bv2 - w2

Examples of Network DynamicsCoupled Fitzhugh-Nagumo Equations for 2 neurons

dv1/dt = v1 (a-v1)(v1-1) - w1 - cv2

dw1/dt = bv1 - w1

Ia = 0

dv2/dt = v2 (a-v2)(v2-1) - w2 - cv1

dw2/dt = bv2 - w2

Examples of Network DynamicsCoupled Fitzhugh-Nagumo Equations for 2 neurons

Identical waveforms — half-period phase difference

a = b = = 0.5

c = 1.1

Examples of Network Dynamics3-cell bidirectional ring

Examples of Network Dynamics3-cell bidirectional ring

Identical waveforms — 1/3-period phase difference

Examples of Network Dynamics3-cell bidirectional ring

2 cells have identical waveforms — half-period phase difference. Third cell has double frequency.

Synchrony and Phase Patterns

Spatial symmetry

distinct cells are synchronous

Temporal symmetry

cell state is time-periodic

Spatio-temporal symmetry

distinct cells are identical except for phase shift

Multirhythms

a cell is identical to itself with a nontrivial phase shift — rational frequency relationships. This is a special type of resonance caused by symmetry

Example: Animal GaitsExample: Animal Gaits

Eadweard Muybridge

Common Animal Gaits

WALKWALK

LEFT rearLEFT rear

LEFT frontLEFT front

RIGHT rearRIGHT rear

RIGHT frontRIGHT front

TROTTROT

LEFT rear + RIGHT frontLEFT rear + RIGHT front

RIGHT rear + LEFT frontRIGHT rear + LEFT front

Common Animal GaitsCommon Animal Gaits

CANTERCANTER

LEFT rear LEFT rear

RIGHT rear + LEFT frontRIGHT rear + LEFT front

RIGHT frontRIGHT front

Common Animal GaitsCommon Animal Gaits

TRANSVERSE GALLOPTRANSVERSE GALLOP

LEFT rear + (delay) RIGHT LEFT rear + (delay) RIGHT rearrear

LEFT front + (delay) RIGHT LEFT front + (delay) RIGHT frontfront

Common Animal GaitsCommon Animal Gaits

RACK or PACERACK or PACE

LEFT rear + LEFT frontLEFT rear + LEFT front

RIGHT rear + RIGHT frontRIGHT rear + RIGHT front

Common Animal GaitsCommon Animal Gaits

Pattern of PhasesPattern of Phases

WALKWALK

Four legs hit the ground Four legs hit the ground at equally spaced times at equally spaced times — from back to front: — from back to front: left, then rightleft, then right

0.750.50.250

Pattern of PhasesPattern of Phases

0.50.500

TROTTROT

Diagonal pairs of legs hit Diagonal pairs of legs hit the ground alternatelythe ground alternately

Pattern of PhasesPattern of Phases

PACEPACE

Left legs hit the ground Left legs hit the ground together; then right legs together; then right legs hit the ground togetherhit the ground together

0.50.500

Pattern of PhasesPattern of Phases

BOUNDBOUND

Rear legs hit the ground Rear legs hit the ground together; then front legs together; then front legs hit the ground togetherhit the ground together

0.500.50

Pattern of PhasesPattern of Phases

TRANSVERSE TRANSVERSE GALLOPGALLOP

Like a bound but with Like a bound but with slight delays in each pair slight delays in each pair of legsof legs

0.60.10.50

Pattern of PhasesPattern of Phases

ROTARY ROTARY GALLOPGALLOP

Like transverse gallop Like transverse gallop but one pair of legs uses but one pair of legs uses opposite timingopposite timing

0.50.10.60

Pattern of PhasesPattern of Phases

CANTERCANTER

One diagonal pair of legs One diagonal pair of legs is synchronised; other is synchronised; other pair alternatespair alternates

0.50.20.20

Pattern of PhasesPattern of Phases

PRONKPRONK

0000

Pronk ?Pronk ?

Pronking AlpacaPronking Alpaca

Pattern of PhasesPattern of Phases

PRONKPRONK

0000

00000.50.80.80

0.750.50.250

0.60.10.50 0.50.10.60

0.50.5000.50.500 0.500.50

WALKWALK TROTTROT PACEPACE BOUNDBOUND

TRANSVERSE TRANSVERSE GALLOPGALLOP

ROTARY ROTARY GALLOPGALLOP CANTERCANTER PRONKPRONK

00000.50.80.80

0.750.50.250

0.60.10.50 0.50.10.60

0.50.500

0.50.500

0.500.50

WHYWHY are there so are there so many?many?

HOWHOW are they are they produced?produced?

00000.50.80.80

0.750.50.250

0.60.10.50 0.50.10.60

0.50.500

0.50.500

0.500.50

WHYWHY are there so are there so many?many?

00000.50.80.80

0.750.50.250

0.60.10.50 0.50.10.60

0.50.500

0.50.500

0.500.50

WHYWHY are there so are there so many?many?

EFFICIENCYEFFICIENCY and and EFFECTIVENESSEFFECTIVENESS

00000.50.80.80

0.750.50.250

0.60.10.50 0.50.10.60

0.50.500

0.50.500

0.500.50

HOWHOW are they are they produced?produced?

00000.50.80.80

0.750.50.250

0.60.10.50 0.50.10.60

0.50.500

0.50.500

0.500.50

CCENTRALENTRAL PPATTERNATTERN GGENERATORENERATOR

HOWHOW are they are they produced?produced?

Central Pattern GeneratorCentral Pattern Generator

Network of Network of nerve cells nerve cells ((neuronsneurons) in ) in the spinal the spinal column, column, notnot in the brainin the brain

Coupled OscillatorsCoupled Oscillators

Coupled OscillatorsCoupled Oscillators

In phaseIn phase Out of phaseOut of phase

same statesame stateat all timesat all times

state lags bystate lags byhalf the periodhalf the period

Coupled OscillatorsCoupled Oscillators

Coupled OscillatorsCoupled Oscillators

Coupled OscillatorsCoupled Oscillators

Coupled OscillatorsCoupled Oscillators

Coupled OscillatorsCoupled Oscillators

0000 00.500.5

00.750.50.25 00.250.50.75

Coupled OscillatorsCoupled Oscillators

0000 00.500.5

00.750.50.25 00.250.50.75

pronkpronk trottrot

walkwalk reversereversewalkwalk

But what about the others?But what about the others?

A more detailed model A more detailed model involving the main leg involving the main leg muscle groups uses all muscle groups uses all eight oscillators to eight oscillators to drive the legs: these drive the legs: these four to “push” and the four to “push” and the other four to “pull”other four to “pull”

Four of the oscillators set Four of the oscillators set the pattern of phase shiftsthe pattern of phase shifts

An argument based on An argument based on symmetry suggests an 8-symmetry suggests an 8-oscillator network as the oscillator network as the simplest possibilitysimplest possibility

00.250.50.7500.250.50.75

WALKWALK

00.50.500.5000.5

TROTTROT

00.50.500.5000.5

BOUNDBOUND

PACEPACE

00.50

PRONKPRONK

00000000

ROTARY ROTARY GALLOPGALLOP

TRANSVERSE TRANSVERSE GALLOPGALLOP

CANTERCANTER

These also occur, as “secondary” patternsThese also occur, as “secondary” patterns

00.250.2500.750.50.50.75

BUCKBUCK

Classification of Phase Patterns

H/K Theorem(Buono and Golubitsky)

Let K be the set of all spatial symmetries — those that leave the state of the system unchanged at every instant of time.

Let H be the set of all spatio-temporal symmetries — those that leave the state of the system unchanged except for a phase shift.

Example — the PACEExample — the PACE

PACEPACE

Left legs hit the ground Left legs hit the ground together; then right legs together; then right legs hit the ground togetherhit the ground together

0.50.500

Classification of Phase Patterns

The set K of all spatial symmetries:Leave all legs unchangedSwap front and back

Cyclic group Z2 of order 2.

The set K of all spatio-temporal symmetries:Leave all legs unchangedSwap front and backSwap left and rightSwap front and back and left and right

Dihedral group D2 of order 4.

Here K is normal in H and H/K is cyclic (of order 2).

H/K Theorem

K is normal in H H/K is cyclicplus two more technical conditions

Are necessary and sufficient for H and K to be the spatio-temporal and spatial symmetry groups of some periodic state (for some suitable ODE with the given symmetry)

H/K Theorem

Essentially, the H/K theorem tells us which spatio-temporal symmetries are to be expected.

Other Theorems (such as the Symmetric Hopf Bifurcation Theorem) provide sufficient conditions for various of these states to occur.

Patterns of Synchrony

The spatial symmetry group K specifies which legs are in synchrony with which.

It divides the legs into synchronous “clusters”

NetwoNetworkrkA A networknetwork or or directed graphdirected graph

consists of a set of:consists of a set of:

•• nodesnodes or or verticesvertices or or cellscells

connected byconnected by

•• directeddirected edgesedges or or arrowsarrows

NetwoNetworkrkEach cell has a Each cell has a cell-typecell-type and each arrow has an and each arrow has an arrow-typearrow-type, allowing us , allowing us to require the cells or to require the cells or arrows concerned to arrows concerned to have ‘the same’ have ‘the same’ structure. In effect these structure. In effect these are are labelslabels on the cells on the cells and arrows. Abstractly and arrows. Abstractly they are specified by they are specified by equivalence relationsequivalence relations on on the set of cells and the the set of cells and the set of arrows.set of arrows.

NetwoNetworkrk Arrows may form Arrows may form loopsloops (same head and tail), (same head and tail),

and there may be and there may be multiple arrowsmultiple arrows (connecting (connecting the same pair of cells).the same pair of cells).

Special case: Special case: regular homogeneous networksregular homogeneous networks. .

These have one type of cell, one type of arrow, These have one type of cell, one type of arrow, and the number of arrows entering each cell is and the number of arrows entering each cell is the same.the same.

This number is the This number is the valencyvalency of the network. of the network.

Regular Homogeneous Regular Homogeneous NetworkNetworkThis is a regular homogeneous network of valency 3. This is a regular homogeneous network of valency 3.

12345

Network EnumerationNetwork EnumerationNN vv=1=1 vv=2=2 vv=3=3 vv=4=4 vv=5=5 vv=6=6

11 11 11 11 11 11 11

22 33 66 1010 1515 2121 2828

33 77 4444 180180 590590 15821582 37243724

44 1919 475475 69156915 6342063420 412230412230 20808272080827

55 4747 68746874 444722444722 104072268104072268 265076184265076184 34056654123405665412

66 130130 126750126750 4324260443242604 55696772105569677210 355906501686355906501686 1350853483470413508534834704

Number of topologically distinct regular homogeneous networks on N cells with valency v

Network Network DynamicsDynamics

To any network we associate a class of To any network we associate a class of admissible vector fieldsadmissible vector fields, defining , defining admissible admissible ODEsODEs, which consists of those vector fields, which consists of those vector fields

FF((xx))

That respect the network structure, and the That respect the network structure, and the corresponding ODEscorresponding ODEs

ddxx/d/dtt = = FF((xx))

What does ‘respect the network structure’ mean?

Admissible Admissible ODEsODEs

Admissible ODEsAdmissible ODEs are are defined in terms of the defined in terms of the input structureinput structure of the of the network.network.

The The input setinput set II((cc)) of a cell of a cell cc is the set of all arrows is the set of all arrows whose head is whose head is cc..

This This includesincludes multiple multiple arrows and loops. arrows and loops.

Admissible Admissible ODEsODEs

Choose coordinates Choose coordinates xxcc R Rkk for each cell for each cell cc. .

(We use (We use RRkk for simplicity, and because we for simplicity, and because we consider only consider only locallocal bifurcation). Then bifurcation). Then

ddxxcc/d/dtt = = ff((xxcc,,xxTT((II ( (cc))))))

where where TT((II((cc)))) is the tuple of tail cells of is the tuple of tail cells of II((cc))..

Admissible Admissible ODEsODEs

ddxx11/d/dtt = = ff((xx11,,xx11, , xx22, , xx33, , xx33, , xx44, , xx55, , xx55, , xx55))

12345

ddxxcc/d/dtt = = ff((xxcc,,xxTT((II ( (cc))))))

Admissible Admissible ODEsODEsAdmissible ODEs for the example network:Admissible ODEs for the example network:

ddxx11/d/dtt = = ff((xx11,, xx22,, xx22,, xx33) )

ddxx22/d/dtt = = ff((xx22,, xx33,, xx44,, xx55) )

ddxx33/d/dtt = = ff((xx33,, xx11,, xx33,, xx44) )

ddxx44/d/dtt = = ff((xx44,, xx22,, xx33,, xx55) )

ddxx55/d/dtt = = ff((xx55,, xx22,, xx44,, xx44))Where Where ff satisfies the symmetry conditionsatisfies the symmetry condition

ff((xx,,uu,,vv,,ww)) is symmetric in is symmetric in uu, , vv, , ww

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Admissible Admissible ODEsODEsBecause the network is regular and homogeneous, Because the network is regular and homogeneous,

the condition “respect the network structure” implies the condition “respect the network structure” implies that in any admissible ODEthat in any admissible ODE

ddxxcc/d/dtt = = ff((xxcc,,xxTT((II ( (cc))))))

the the samesame function function ff occurs in each equation. occurs in each equation.

Moreover, Moreover, ff is is symmetricsymmetric in the variables in the variables xxTT((II ( (cc))))..

However, the first variable is distinguished, so However, the first variable is distinguished, so ff is is not required to be symmetric in that variable.not required to be symmetric in that variable.

AdmissibleAdmissible ODEs are ODEs are those whose structure those whose structure reflects the network reflects the network topology and the types of topology and the types of the cells and arrowsthe cells and arrows

dxdx11/dt = f/dt = f11((xx11,x,x22,x,x44,x,x55))

dxdx22/dt = f/dt = f22((xx22,x,x11,x,x33,x,x55))

dxdx33/dt = f/dt = f33((xx33,x,x11,x,x44))

dxdx44/dt = f/dt = f44((xx44,x,x22,x,x44))

dxdx55/dt = f/dt = f55((xx55,x,x44))12453

domain conditiondomain condition

dxdx11/dt = f(/dt = f(xx11,x,x22,x,x44,x,x55))

dxdx22/dt = f(/dt = f(xx22,x,x11,x,x33,x,x55))

dxdx33/dt = g(/dt = g(xx33,x,x11,x,x44))

dxdx44/dt = g(/dt = g(xx44,x,x22,x,x44))

dxdx55/dt = h(/dt = h(xx55,x,x44))12453

pullback conditionpullback condition

dxdx11/dt = f(/dt = f(xx11,,xx22,,xx44,,xx55))

dxdx22/dt = f(/dt = f(xx22,,xx11,,xx33,,xx55))

dxdx33/dt = g(/dt = g(xx33,,xx11,,xx44))

dxdx44/dt = g(/dt = g(xx44,,xx22,,xx44))

dxdx55/dt = h(/dt = h(xx55,,xx44))12453

Vertex groupVertex group symmetry symmetry

dxdx11/dt = f(/dt = f(xx11,,xx22,,xx44,,xx55))

dxdx22/dt = f(/dt = f(xx22,,xx11,,xx33,,xx55))

dxdx33/dt = g(/dt = g(xx33,,xx11,,xx44))

dxdx44/dt = g(/dt = g(xx44,,xx22,,xx44))

dxdx55/dt = h(/dt = h(xx55,,xx44))12453

How do synchronous states How do synchronous states behave?behave?

[with M.Golubitsky and M.Pivato] Symmetry groupoids and [with M.Golubitsky and M.Pivato] Symmetry groupoids and patterns of synchrony in coupled cell networks, patterns of synchrony in coupled cell networks, SIAM J. SIAM J. Appl. Dyn. Sys.Appl. Dyn. Sys. 22 (2003) 609-646. DOI: (2003) 609-646. DOI: 10.1137/S111111110341989610.1137/S1111111103419896

[with M.Golubitsky and M.Nicol] Some curious phenomena [with M.Golubitsky and M.Nicol] Some curious phenomena in coupled cell networks, in coupled cell networks, J. Nonlin. SciJ. Nonlin. Sci. . 1414 (2004) 207- (2004) 207-236236..

[with M.Golubitsky and A.Török] Patterns of synchrony in [with M.Golubitsky and A.Török] Patterns of synchrony in coupled cell networks with multiple arrows, coupled cell networks with multiple arrows, SIAM J. Appl. SIAM J. Appl. Dyn. Sys.Dyn. Sys. 44 (2005) 78-100. [DOI: 10.1137/040612634] (2005) 78-100. [DOI: 10.1137/040612634]

[With F.A.M.Aldosray] Enumeration of homogeneous [With F.A.M.Aldosray] Enumeration of homogeneous coupled cell networks, coupled cell networks, Internat. J. Bif. ChaosInternat. J. Bif. Chaos 1515 (2005) (2005) 2361-2373.2361-2373.

[with M.Golubitsky] Nonlinear dynamics of networks: the [with M.Golubitsky] Nonlinear dynamics of networks: the groupoid formalism, groupoid formalism, Bull. Amer. Math. SocBull. Amer. Math. Soc. . 4343 (2006) (2006) 305-364.305-364.

ReferencesReferences

Dynamics of Networksto be continued...

Ian Stewart

Mathematics Institute

University of Warwick

UK-Japan Winter SchoolDynamics and Complexity