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Transcript of Dynamics of Networks 1 Basic Formalism & Symmetry Ian Stewart Mathematics Institute University of...
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
12345
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