text: [4] P. C. W. Davies. A quantum theory of Wheeler-Feynman electrodynamics. Mathe- matical Proceedings of the Cambridge Philosophical Society, 68(3):751–764, 1970. [5] P. C. W. Davies. Extension of Wheeler-Feynman quantum theory to the relativistic domain. I. Scattering processes. J. Phys. A: Math. Theor., 4:836–845, 1971. [6] P. C. W. Davies. Extension of Wheeler-Feynman quantum theory to the relativistic domain. II. Emission processes. J. of Phys. A: Math. Theor., 5(7):1025, 1972. [7] F. Hoyle and J. V. Narlikar. Cosmology and action-at-a-distance electrodynamics. Rev. Mod. Phys., 67:113–155, 1995. [8] A. O. Barut. The covariant many-body problem in quantumelectrodynamics. J. Math. Phys., 32:1091–1095, 1991. [9] D.-A. Deckert. Electrodynamic Absorber Theory - A Mathematical Study. PhD thesis, Mathematisches Institut, Ludwig-Maximilians-Universität München, 2009. http:// edoc.ub.uni-muenchen.de/11421/. [10] P. A. M. Dirac. Relativistic Quantum Mechanics. Proc. R. Soc. Lond. A, 136:453–464, 1932. [11] S. Tomonaga. On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. Prog. Theor. Phys., 1:27–42, 1946. [12] J. Schwinger. Quantum Electrodynamics. I. A Covariant Formulation. Phys. Rev., 74(2162):1439–1461, 1948. [13] S. Petrat and R. Tumulka. Multi-Time Schrödinger Equations Cannot Contain Inter- action Potentials. J. Math. Phys., 55(032302), 2014. https://arxiv.org/abs/1308. 1065. [14] S. Petrat and R. Tumulka. Multi-Time Wave Functions for Quantum Field Theory. Ann. Phys., 345:17–54, 2014. https://arxiv.org/abs/1309.0802v3. [15] S. Petrat and R. Tumulka. Multi-time formulation of pair creation. J. Phys. A: Math. Theor., 47(11):112001, 2014. [16] M. Lienert. A relativistically interacting exactly solvable multi-time model for two massless Dirac particles in 1+1 dimensions. J. Math. Phys., 56(4):042301, 2015. https://arxiv.org/abs/1411.2833. [17] M. Lienert and L. Nickel. A simple explicitly solvable interacting relativistic N-particle model. J. Phys. A: Math. Theor., 48(32):325301, 2015. https://arxiv.org/abs/ 1502.00917. [18] M. Lienert. PhD thesis. Lorentz invariant quantum dynamics in the multi-time for- malism, 2015. Mathematical Institute, Ludwig-Maximilians-Universität, Munich, Ger- many. https://edoc.ub.uni-muenchen.de/18705/. [19] M. Lienert. On the question of current conservation for the Two-Body Dirac equations of constraint theory. J. Phys. A: Math. Theor., 48(32):325302, 2015. https://arxiv. org/abs/1501.07027. [20] D.-A. Deckert and L. Nickel. Consistency of multi-time Dirac equations with general interaction potentials. J. Math. Phys., 57(7):072301, 2016. https://arxiv.org/abs/ 1603.02538. 31

text: P.

text: C.

text: W.

text: Davies.

text: A

text: quantum

text: theory

text: of

text: Wheeler-Feynman

text: electrodynamics.

text: Mathe-

text: matical

text: Proceedings

text: of

text: the

text: Cambridge

text: Philosophical

text: Society,

text: P.

text: C.

text: W.

text: Davies.

text: Extension

text: of

text: Wheeler-Feynman

text: quantum

text: theory

text: to

text: the

text: relativistic

text: domain.

text: I.

text: Scattering

text: processes.

text: J.

text: Phys.

text: A:

text: Math.

text: Theor.,

text: P.

text: C.

text: W.

text: Davies.

text: Extension

text: of

text: Wheeler-Feynman

text: quantum

text: theory

text: to

text: the

text: relativistic

text: domain.

text: II.

text: Emission

text: processes.

text: J.

text: of

text: Phys.

text: A:

text: Math.

text: Theor.,

text: F.

text: Hoyle

text: and

text: J.

text: V.

text: Narlikar.

text: Cosmology

text: and

text: action-at-a-distance

text: electrodynamics.

text: Rev.

text: Mod.

text: Phys.,

text: A.

text: O.

text: Barut.

text: The

text: covariant

text: many-body

text: problem

text: in

text: quantumelectrodynamics.

text: J.

text: Math.

text: Phys.,

text: D.-A.

text: Deckert.

text: Electrodynamic

text: Absorber

text: Theory

text: A

text: Mathematical

text: Study.

text: PhD

text: thesis,

text: Mathematisches

text: Institut,

text: Ludwig-Maximilians-Universität

text: München,

text: P.

text: A.

text: M.

text: Dirac.

text: Relativistic

text: Quantum

text: Mechanics.

text: Proc.

text: R.

text: Soc.

text: Lond.

text: A,

text: S.

text: Tomonaga.

text: On

text: a

text: Relativistically

text: Invariant

text: Formulation

text: of

text: the

text: Quantum

text: Theory

text: of

text: Wave

text: Fields.

text: Prog.

text: Theor.

text: Phys.,

text: J.

text: Schwinger.

text: Quantum

text: Electrodynamics.

text: I.

text: A

text: Covariant

text: Formulation.

text: Phys.

text: Rev.,

text: S.

text: Petrat

text: and

text: R.

text: Tumulka.

text: Multi-Time

text: Schrödinger

text: Equations

text: Cannot

text: Contain

text: Inter-

text: action

text: Potentials.

text: J.

text: Math.

text: Phys.,

text: S.

text: Petrat

text: and

text: R.

text: Tumulka.

text: Multi-Time

text: Wave

text: Functions

text: for

text: Quantum

text: Field

text: Theory.

text: Ann.

text: Phys.,

text: S.

text: Petrat

text: and

text: R.

text: Tumulka.

text: Multi-time

text: formulation

text: of

text: pair

text: creation.

text: J.

text: Phys.

text: A:

text: Math.

text: Theor.,

text: M.

text: Lienert.

text: A

text: relativistically

text: interacting

text: exactly

text: solvable

text: multi-time

text: model

text: for

text: two

text: massless

text: Dirac

text: particles

text: in

text: dimensions.

text: J.

text: Math.

text: Phys.,

text: M.

text: Lienert

text: and

text: L.

text: Nickel.

text: A

text: simple

text: explicitly

text: solvable

text: interacting

text: relativistic

text: N-particle

text: model.

text: J.

text: Phys.

text: A:

text: Math.

text: Theor.,

text: M.

text: Lienert.

text: PhD

text: thesis.

text: Lorentz

text: invariant

text: quantum

text: dynamics

text: in

text: the

text: multi-time

text: for-

text: malism,

text: Mathematical

text: Institute,

text: Ludwig-Maximilians-Universität,

text: Munich,

text: Ger-

text: many.

text: M.

text: Lienert.

text: On

text: the

text: question

text: of

text: current

text: conservation

text: for

text: the

text: Two-Body

text: Dirac

text: equations

text: of

text: constraint

text: theory.

text: J.

text: Phys.

text: A:

text: Math.

text: Theor.,

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text: Deckert

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text: Math.

text: Phys.,


text: similarly to the homeostatic balance of inhibitory synaptic plasticity as previously suggested (Vo- gels et al., 2011; Luz and Shamir, 2012). Yet, in our model, inhibitory plasticity plays a central role in multi-area, deep error coding, which goes beyond the standard view of inhibitory plasticity as a homeostatic stabilizing force (Keck et al., 2017). Error minimization is an integral part of brain function according to predictive coding theories (Rao and Ballard, 1999; Friston, 2005), and backprop can be mapped onto a predictive coding network architecture (Whittington and Bogacz, 2017). From a formal point of view this approach is encompassed by the framework introduced by LeCun (1988). A possible network implementation is suggested by Whittington and Bogacz (2017) that requires intricate circuitry with appropriately tuned error-representing neurons. According to that model, the only plastic synapses are those that connect prediction and error neurons. We built upon the previously made observation that top-down and bottom-up weights need not be in perfect agreement to enable multi-area error-driven learning (Lee et al., 2015; Lillicrap et al., 2016). Consistent with these ﬁndings, the strict weight symmetry arising in the classical error backpropagation algorithm is not required in our case either for a successful learning in hidden area neurons. We have also shown that top-down synapses can be learned using the same dendritic predictive learning rule used at the remaining connections. In our model, the top-down connections have a dual role: they are involved in the apical error representation and, they learn to match the somatic ﬁring driven by the bottom-up input (Urbanczik and Senn, 2014). The simultaneous learning of the bottom-up and top-down pathways leads to the formation of a generative network that can denoise sensory input or generate dream-like inputs (Fig. 5). Finally, the framework introduced here could also be adapted to other types of error-based learn- ing, such as in generative models that instead of learning to discriminate sensory inputs, learn to generate following sensory input statistics. Error propagation in these forms of generative models, which arise from an inaccurate prediction of sensory inputs, may rely on different dendritic com- partments and interneurons, such as the previously mentioned PV inhibitory cells (Petreanu et al., 2009). Acknowledgements The authors would like to thank Timothy P. Lillicrap, Blake Richards, Benjamin Scellier and Mihai 16

text: similarly

label: paragraph

font: QRCTXG+NimbusSanL-Regu

text: to

text: the

text: homeostatic

text: balance

text: of

text: inhibitory

text: synaptic

text: plasticity

text: as

text: previously

text: suggested

text: (Vo-

text: gels

text: et

text: al.,

text: Luz

text: and

text: Shamir,

text: Yet,

text: in

text: our

text: model,

text: inhibitory

text: plasticity

text: plays

text: a

text: central

text: role

text: in

text: multi-area,

text: deep

text: error

text: coding,

text: which

text: goes

text: beyond

text: the

text: standard

text: view

text: of

text: inhibitory

text: plasticity

text: as

text: a

text: homeostatic

text: stabilizing

text: force

text: (Keck

text: et

text: al.,

text: Error

text: minimization

text: is

text: an

text: integral

text: part

text: of

text: brain

text: function

text: according

text: to

text: predictive

text: coding

text: theories

text: (Rao

text: and

text: Ballard,

text: Friston,

text: and

text: backprop

text: can

text: be

text: mapped

text: onto

text: a

text: predictive

text: coding

text: network

text: architecture

text: (Whittington

text: and

text: Bogacz,

text: From

text: a

text: formal

text: point

text: of

text: view

text: this

text: approach

text: is

text: encompassed

text: by

text: the

text: framework

text: introduced

text: by

text: LeCun

text: A

text: possible

text: network

text: implementation

text: is

text: suggested

text: by

text: Whittington

text: and

text: Bogacz

text: that

text: requires

text: intricate

text: circuitry

text: with

text: appropriately

text: tuned

text: error-representing

text: neurons.

text: According

text: to

text: that

text: model,

text: the

text: only

text: plastic

text: synapses

text: are

text: those

text: that

text: connect

text: prediction

text: and

text: error

text: neurons.

text: We

text: built

text: upon

text: the

text: previously

text: made

text: observation

text: that

text: top-down

text: and

text: bottom-up

text: weights

text: need

text: not

text: be

text: in

text: perfect

text: agreement

text: to

text: enable

text: multi-area

text: error-driven

text: learning

text: (Lee

text: et

text: al.,

text: Lillicrap

text: et

text: al.,

text: Consistent

text: with

text: these

text: ﬁndings,

text: the

text: strict

text: weight

text: symmetry

text: arising

text: in

text: the

text: classical

text: error

text: backpropagation

text: algorithm

text: is

text: not

text: required

text: in

text: our

text: case

text: either

text: for

text: a

text: successful

text: learning

text: in

text: hidden

text: area

text: neurons.

text: We

text: have

text: also

text: shown

text: that

text: top-down

text: synapses

text: can

text: be

text: learned

text: using

text: the

text: same

text: dendritic

text: predictive

text: learning

text: rule

text: used

text: at

text: the

text: remaining

text: connections.

text: In

text: our

text: model,

text: the

text: top-down

text: connections

text: have

text: a

text: dual

text: role:

text: they

text: are

text: involved

text: in

text: the

text: apical

text: error

text: representation

text: and,

text: they

text: learn

text: to

text: match

text: the

text: somatic

text: ﬁring

text: driven

text: by

text: the

text: bottom-up

text: input

text: (Urbanczik

text: and

text: Senn,

text: The

text: simultaneous

text: learning

text: of

text: the

text: bottom-up

text: and

text: top-down

text: pathways

text: leads

text: to

text: the

text: formation

text: of

text: a

text: generative

text: network

text: that

text: can

text: denoise

text: sensory

text: input

text: or

text: generate

text: dream-like

text: inputs

text: (Fig.

text: Finally,

text: the

text: framework

text: introduced

text: here

text: could

text: also

text: be

text: adapted

text: to

text: other

text: types

text: of

text: error-based

text: learn-

text: ing,

text: such

text: as

text: in

text: generative

text: models

text: that

text: instead

text: of

text: learning

text: to

text: discriminate

text: sensory

text: inputs,

text: learn

text: to

text: generate

text: following

text: sensory

text: input

text: statistics.

text: Error

text: propagation

text: in

text: these

text: forms

text: of

text: generative

text: models,

text: which

text: arise

text: from

text: an

text: inaccurate

text: prediction

text: of

text: sensory

text: inputs,

text: may

text: rely

text: on

text: different

text: dendritic

text: com-

text: partments

text: and

text: interneurons,

text: such

text: as

text: the

text: previously

text: mentioned

text: PV

text: inhibitory

text: cells

text: (Petreanu

text: et

text: al.,

text: Acknowledgements

text: The

text: authors

text: would

text: like

text: to

text: thank

text: Timothy

text: P.

text: Lillicrap,

text: Blake

text: Richards,

text: Benjamin

text: Scellier

text: and

text: Mihai


text: − distribution U( 1, 1). We choose a soft rectifying nonlinearity as the neuronal transfer function, φ(u) = log(1 + exp(u)). The pyramidal neuron network is initialized to a self-predicting state where WIP = WPP and 1,1 2,1 − WPI = WPP. Top down weight matrix WPP is ﬁxed and set at random with entries drawn from 1,1 1,2 1,2 a uniform distribution. Output area teaching currents iP are set so as to nudge uP towards the 2 2 teacher-generated utrgt. Reported error curves are exponential moving averages of the sum of 2 (cid:107) − (cid:107) squared errors loss rP rtrgt 2 computed after every example on unseen input patterns. Plas- 2 2 ticity induction terms given by the right-hand sides of Eqs. 7-9 are low-pass ﬁltered with time con- stant τ before being deﬁnitely consolidated, to dampen ﬂuctuations; synaptic plasticity is kept on w throughout. Plasticity and neuron model parameters are given in the accompanying supplementary material. MNIST image classiﬁcation and reconstruction tasks. When simulating the larger models used on the MNIST data set we resort to a discrete-time network dynamics where the compartmen- tal potentials are updated in two steps before applying synaptic changes. The simpliﬁed model dynamics is as follows. For each presented MNIST image, both pyramidal and interneurons are ﬁrst initialized to their bottom-up prediction state (3), u = v , starting from k B,k area 1 upto top area N. Output area neurons are then nudged towards their desired target utrgt, N − yielding updated somatic potentials uP = (1 λ ) v + λ utrgt. To obtain the remaining ﬁnal N N B,N N N − compartmental potentials, the network is revisited in reverse order, proceeding from area k = N 1 down to k = 1. For each k, interneurons are ﬁrst updated to include top-down teaching signals, − uI = (1 λ ) vI + λ uP ; this yields apical compartment potentials according to (4), after which k I k I k+1 we update hidden area somatic potentials as a convex combination with mixing factor λ . The k convex combination factors introduced above are directly related to neuron model parameters as conductance ratios. Synaptic weights are then updated according to Eqs. 7-10. Such simpliﬁed dynamics approximates the full recurrent network relaxation in the deterministic → setting σ 0, with the approximation improving as the top-down dendritic coupling is decreased, → g 0. A We train the models on the standard MNIST handwritten image database, further splitting the training set into 55000 training and 5000 validation examples. The reported test error curves are computed on the 10000 held-aside test images. The four-area network shown in Fig. 4 is initialized in a self-predicting state with appropriately scaled initial weight matrices. To speed-up training we 21

text: distribution

text: U(

font: UFMVCZ+NimbusSanL-ReguItal

text: We

text: choose

text: a

text: soft

text: rectifying

text: nonlinearity

text: as

text: the

text: neuronal

text: transfer

text: function,

text: φ(u)

text: exp(u)).

text: The

text: pyramidal

text: neuron

text: network

text: is

text: initialized

text: to

text: a

text: self-predicting

text: state

text: where

text: WIP

text: WPP

text: and

text: WPI

text: WPP.

text: Top

text: down

text: weight

text: matrix

text: WPP

text: is

text: ﬁxed

text: and

text: set

text: at

text: random

text: with

text: entries

text: drawn

text: from

text: a

text: uniform

text: distribution.

text: Output

text: area

text: teaching

text: currents

text: iP

text: are

text: set

text: so

text: as

text: to

text: nudge

text: uP

text: towards

text: the

text: teacher-generated

text: utrgt.

text: Reported

text: error

text: curves

text: are

text: exponential

text: moving

text: averages

text: of

text: the

text: sum

text: of

text: squared

text: errors

text: loss

text: rP

text: rtrgt

text: computed

text: after

text: every

text: example

text: on

text: unseen

text: input

text: patterns.

text: Plas-

text: ticity

text: induction

text: terms

text: given

text: by

text: the

text: right-hand

text: sides

text: of

text: Eqs.

text: are

text: low-pass

text: ﬁltered

text: with

text: time

text: con-

text: stant

text: τ

text: before

text: being

text: deﬁnitely

text: consolidated,

text: to

text: dampen

text: ﬂuctuations;

text: synaptic

text: plasticity

text: is

text: kept

text: on

text: w

text: throughout.

text: Plasticity

text: and

text: neuron

text: model

text: parameters

text: are

text: given

text: in

text: the

text: accompanying

text: supplementary

text: material.

text: MNIST

text: image

text: classiﬁcation

text: and

text: reconstruction

text: tasks.

text: When

text: simulating

text: the

text: larger

text: models

text: used

text: on

text: the

text: MNIST

text: data

text: set

text: we

text: resort

text: to

text: a

text: discrete-time

text: network

text: dynamics

text: where

text: the

text: compartmen-

text: tal

text: potentials

text: are

text: updated

text: in

text: two

text: steps

text: before

text: applying

text: synaptic

text: changes.

text: The

text: simpliﬁed

text: model

text: dynamics

text: is

text: as

text: follows.

text: For

text: each

text: presented

text: MNIST

text: image,

text: both

text: pyramidal

text: and

text: interneurons

text: are

text: ﬁrst

text: initialized

text: to

text: their

text: bottom-up

text: prediction

text: state

text: u

text: v

text: starting

text: from

text: k

text: B,k

text: area

text: upto

text: top

text: area

text: N.

text: Output

text: area

text: neurons

text: are

text: then

text: nudged

text: towards

text: their

text: desired

text: target

text: utrgt,

text: N

text: yielding

text: updated

text: somatic

text: potentials

text: uP

text: λ

text: v

text: λ

text: utrgt.

text: To

text: obtain

text: the

text: remaining

text: ﬁnal

text: N

text: N

text: B,N

text: N

text: N

text: compartmental

text: potentials,

text: the

text: network

text: is

text: revisited

text: in

text: reverse

text: order,

text: proceeding

text: from

text: area

text: k

text: N

text: down

text: to

text: k

text: For

text: each

text: k,

text: interneurons

text: are

text: ﬁrst

text: updated

text: to

text: include

text: top-down

text: teaching

text: signals,

text: uI

text: λ

text: vI

text: λ

text: uP

text: this

text: yields

text: apical

text: compartment

text: potentials

text: according

text: to

text: after

text: which

text: k

text: I

text: k

text: I

text: we

text: update

text: hidden

text: area

text: somatic

text: potentials

text: as

text: a

text: convex

text: combination

text: with

text: mixing

text: factor

text: λ

text: The

text: k

text: convex

text: combination

text: factors

text: introduced

text: above

text: are

text: directly

text: related

text: to

text: neuron

text: model

text: parameters

text: as

text: conductance

text: ratios.

text: Synaptic

text: weights

text: are

text: then

text: updated

text: according

text: to

text: Eqs.

text: Such

text: simpliﬁed

text: dynamics

text: approximates

text: the

text: full

text: recurrent

text: network

text: relaxation

text: in

text: the

text: deterministic

text: setting

text: σ

text: with

text: the

text: approximation

text: improving

text: as

text: the

text: top-down

text: dendritic

text: coupling

text: is

text: decreased,

text: g

text: A

text: We

text: train

text: the

text: models

text: on

text: the

text: standard

text: MNIST

text: handwritten

text: image

text: database,

text: further

text: splitting

text: the

text: training

text: set

text: into

text: training

text: and

text: validation

text: examples.

text: The

text: reported

text: test

text: error

text: curves

text: are

text: computed

text: on

text: the

text: held-aside

text: test

text: images.

text: The

text: four-area

text: network

text: shown

text: in

text: Fig.

text: is

text: initialized

text: in

text: a

text: self-predicting

text: state

text: with

text: appropriately

text: scaled

text: initial

text: weight

text: matrices.

text: To

text: speed-up

text: training

text: we


text: g g − uI = B vI + som uP = (1 λ) vˆI + λ uP , (S3) k g + g + g k g + g + g k+1 k k+1 lk B som lk B som with g and g the effective dendritic transfer and leak conductances, respectively, and g B lk som the total excitatory and inhibitory teaching conductance. In the equation above, vˆI = gB vI is k g +g k lk B ≡ ∈ the interneuron dendritic prediction (cf. Eq. 8), and λ gsom [0, 1[ is a mixing factor which glk+gB+gsom controls the nudging strength for the interneurons. In other words, the current prediction vˆI and k the teaching signal are averaged with coefﬁcients determined by normalized conductances. We will → later consider the weak nudging limit of λ 0. The relation vˆI = vˆP holds when pyramidal-to-interneuron synaptic weights are equal to k B,k+1 pyramidal-pyramidal forward weights, up to a scale factor: WIP = glk+gB WPP , which simpliﬁes k,k g +g +g k+1,k lk B A to WIP− − = WPP − for the last area where g = 0 (to reduce clutter, we use the slightly abusive N 1,N 1 N,N 1 A notation whereby g should be understood to be zero when referring to output area neurons). This A is the reason for the particular choice of ideal pyramidal-to-interneuron weights presented in the preamble. The network is then internally consistent, in the sense that the interneurons predict the model’s own predictions, held by pyramidal neurons. Bottom-up predictions in the absence of external nudging. We ﬁrst study the situation where the input pattern r is stationary and the output area teaching input is disabled, iP = 0. We 0 N show that the ﬁxed point of the network dynamics is a state where somatic voltages are equal to basal voltages, up to a dendritic attenuation factor. In other words, the network effectively behaves as if it were feedforward, in the sense that it computes the same function as the corresponding network with equal bottom-up but no top-down or lateral connections. − Speciﬁcally, in the absence of external nudging (indicated by the in the superscript), the somatic voltages of pyramidal and interneuron are given by the bottom-up dendritic predictions, − − ≡ g − uP, = vˆP, B WPP− φ(vˆP, − ) (S4) k B,k g + g + g k,k 1 B,k 1 lk B A − − ≡ g − uI, = vˆI, B WIP φ(vˆP, ). (S5) k k g + g k,k B,k lk B To show that Eq. S4 describes the state of the network, we start at the output area and set 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: g

label: equation

text: g

text: uI

font: ZSRHUK+NimbusSanL-Bold

text: B

text: vI

text: som

text: uP

text: λ)

text: vˆI

text: λ

text: uP

text: k

text: g

text: g

text: g

text: k

text: g

text: g

text: g

text: k

text: lk

text: B

text: som

text: lk

text: B

text: som

text: with

text: g

text: and

text: g

text: the

text: effective

text: dendritic

text: transfer

text: and

text: leak

text: conductances,

text: respectively,

text: and

text: g

text: B

text: lk

text: som

text: the

text: total

text: excitatory

text: and

text: inhibitory

text: teaching

text: conductance.

text: In

text: the

text: equation

text: above,

text: vˆI

text: gB

text: vI

text: is

text: k

text: g

text: +g

text: k

text: lk

text: B

text: the

text: interneuron

text: dendritic

text: prediction

text: (cf.

text: Eq.

text: and

text: λ

text: gsom

text: is

text: a

text: mixing

text: factor

text: which

text: glk+gB+gsom

text: controls

text: the

text: nudging

text: strength

text: for

text: the

text: interneurons.

text: In

text: other

text: words,

text: the

text: current

text: prediction

text: vˆI

text: and

text: k

text: the

text: teaching

text: signal

text: are

text: averaged

text: with

text: coefﬁcients

text: determined

text: by

text: normalized

text: conductances.

text: We

text: will

text: later

text: consider

text: the

text: weak

text: nudging

text: limit

text: of

text: λ

text: The

text: relation

text: vˆI

text: vˆP

text: holds

text: when

text: pyramidal-to-interneuron

text: synaptic

text: weights

text: are

text: equal

text: to

text: k

text: pyramidal-pyramidal

text: forward

text: weights,

text: up

text: to

text: a

text: scale

text: factor:

text: WIP

text: glk+gB

text: WPP

text: which

text: simpliﬁes

text: k,k

text: g

text: +g

text: +g

text: lk

text: B

text: A

text: to

text: WIP−

text: WPP

text: for

text: the

text: last

text: area

text: where

text: g

text: (to

text: reduce

text: clutter,

text: we

text: use

text: the

text: slightly

text: abusive

text: N

text: N,N

text: A

text: notation

text: whereby

text: g

text: should

text: be

text: understood

text: to

text: be

text: zero

text: when

text: referring

text: to

text: output

text: area

text: neurons).

text: This

text: A

text: is

text: the

text: reason

text: for

text: the

text: particular

text: choice

text: of

text: ideal

text: pyramidal-to-interneuron

text: weights

text: presented

text: in

text: the

text: preamble.

text: The

text: network

text: is

text: then

text: internally

text: consistent,

text: in

text: the

text: sense

text: that

text: the

text: interneurons

text: predict

text: the

text: model’s

text: own

text: predictions,

text: held

text: by

text: pyramidal

text: neurons.

text: Bottom-up

text: predictions

text: in

text: the

text: absence

text: of

text: external

text: nudging.

text: We

text: ﬁrst

text: study

text: the

text: situation

text: where

text: the

text: input

text: pattern

text: r

text: is

text: stationary

text: and

text: the

text: output

text: area

text: teaching

text: input

text: is

text: disabled,

text: iP

text: We

text: N

text: show

text: that

text: the

text: ﬁxed

text: point

text: of

text: the

text: network

text: dynamics

text: is

text: a

text: state

text: where

text: somatic

text: voltages

text: are

text: equal

text: to

text: basal

text: voltages,

text: up

text: to

text: a

text: dendritic

text: attenuation

text: factor.

text: In

text: other

text: words,

text: the

text: network

text: effectively

text: behaves

text: as

text: if

text: it

text: were

text: feedforward,

text: in

text: the

text: sense

text: that

text: it

text: computes

text: the

text: same

text: function

text: as

text: the

text: corresponding

text: network

text: with

text: equal

text: bottom-up

text: but

text: no

text: top-down

text: or

text: lateral

text: connections.

text: Speciﬁcally,

text: in

text: the

text: absence

text: of

text: external

text: nudging

text: (indicated

text: by

text: the

text: in

text: the

text: superscript),

text: the

text: somatic

text: voltages

text: of

text: pyramidal

text: and

text: interneuron

text: are

text: given

text: by

text: the

text: bottom-up

text: dendritic

text: predictions,

text: g

text: uP,

text: vˆP,

text: B

text: WPP−

text: φ(vˆP,

text: k

text: B,k

text: g

text: g

text: g

text: k,k

text: B,k

text: lk

text: B

text: A

text: g

text: uI,

text: vˆI,

text: B

text: WIP

text: φ(vˆP,

text: k

text: k

text: g

text: g

text: k,k

text: B,k

text: lk

text: B

text: To

text: show

text: that

text: Eq.

text: describes

text: the

text: state

text: of

text: the

text: network,

text: we

text: start

text: at

text: the

text: output

text: area

text: and

text: set

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: terneuron ﬁring rates. Such deviation can be intuitively understood as an area-wise interneuron prediction mismatch, being zero when interneurons perfectly explain pyramidal neuron activity. We now evaluate this difference vector at a ﬁxed point to obtain a recurrence relation that links consec- utive areas. The steady-state somatic potentials of hidden pyramidal neurons are given by g g uP = B vP + A vP = vˆP + λ vP k g + g + g B,k g + g + g A,k B,k A,k lk B A lk B A = vˆP + λ W e . (S7) B,k k,k+1 k+1 To shorten the following, we assumed that the apical attenuation factor is equal to the interneu- ron nudging strength λ. As previously mentioned, we proceed under the assumption of weak feed- back, λ small. As for the corresponding interneurons, we insert Eq. S7 into Eq. S3 and note that when the network is in a self-predicting state we have vˆI − = vˆP , yielding k 1 B,k (cid:0) (cid:1) − uI − = (1 λ) vˆP + λ vˆP + λ vP = vˆP + λ2 vP . (S8) k 1 B,k B,k A,k B,k A,k Using the identities (S7) and (S8), we now expand to ﬁrst order the difference vector e around k vˆP as follows B,k (cid:0) (cid:1) − O (cid:107) (cid:107) e = φ(uP) φ(uI − ) = λ D vP + λ2 vP . (S9) k k k 1 k A,k A,k (cid:48) Matrix D is a diagonal matrix with diagonal equal to φ (vˆP ), i.e., whose i-th element reads k B,k dφ(vˆP ). It contains the derivative of the neuronal transfer function φ evaluated component-wise at dv B,k,i the bottom-up predictions vˆP . Recalling Eq. S6, we obtain a recurrence relation B,k+1 (cid:0) (cid:1) O (cid:107) (cid:107) e = λ D W e + λ2 W e . (S10) k k k,k+1 k+1 k,k+1 k+1 32 ##LTLine## ##LTLine## ##LTLine##

text: terneuron

text: ﬁring

text: rates.

text: Such

text: deviation

text: can

text: be

text: intuitively

text: understood

text: as

text: an

text: area-wise

text: interneuron

text: prediction

text: mismatch,

text: being

text: zero

text: when

text: interneurons

text: perfectly

text: explain

text: pyramidal

text: neuron

text: activity.

text: We

text: now

text: evaluate

text: this

text: difference

text: vector

text: at

text: a

text: ﬁxed

text: point

text: to

text: obtain

text: a

text: recurrence

text: relation

text: that

text: links

text: consec-

text: utive

text: areas.

text: The

text: steady-state

text: somatic

text: potentials

text: of

text: hidden

text: pyramidal

text: neurons

text: are

text: given

text: by

text: g

text: g

text: uP

text: B

text: vP

text: A

text: vP

text: vˆP

text: λ

text: vP

text: k

text: g

text: g

text: g

text: B,k

text: g

text: g

text: g

text: A,k

text: B,k

text: A,k

text: lk

text: B

text: A

text: lk

text: B

text: A

text: vˆP

text: λ

text: W

text: e

text: B,k

text: To

text: shorten

text: the

text: following,

text: we

text: assumed

text: that

text: the

text: apical

text: attenuation

text: factor

text: is

text: equal

text: to

text: the

text: interneu-

text: ron

text: nudging

text: strength

text: λ.

text: As

text: previously

text: mentioned,

text: we

text: proceed

text: under

text: the

text: assumption

text: of

text: weak

text: feed-

text: back,

text: λ

text: small.

text: As

text: for

text: the

text: corresponding

text: interneurons,

text: we

text: insert

text: Eq.

text: into

text: Eq.

text: and

text: note

text: that

text: when

text: the

text: network

text: is

text: in

text: a

text: self-predicting

text: state

text: we

text: have

text: vˆI

text: vˆP

text: yielding

text: k

text: B,k

text: uI

text: λ)

text: vˆP

text: λ

text: vˆP

text: λ

text: vP

text: vˆP

text: λ2

text: vP

text: k

text: B,k

text: B,k

text: A,k

text: B,k

text: A,k

text: Using

text: the

text: identities

text: and

text: we

text: now

text: expand

text: to

text: ﬁrst

text: order

text: the

text: difference

text: vector

text: e

text: around

text: k

text: vˆP

text: as

text: follows

text: B,k

text: O

text: e

text: φ(uP)

text: φ(uI

text: λ

text: D

text: vP

text: λ2

text: vP

text: k

text: k

text: k

text: k

text: A,k

text: A,k

text: Matrix

text: D

text: is

text: a

text: diagonal

text: matrix

text: with

text: diagonal

text: equal

text: to

text: φ

text: (vˆP

text: i.e.,

text: whose

text: i-th

text: element

text: reads

text: k

text: B,k

text: dφ(vˆP

text: It

text: contains

text: the

text: derivative

text: of

text: the

text: neuronal

text: transfer

text: function

text: φ

text: evaluated

text: component-wise

text: at

text: dv

text: B,k,i

text: the

text: bottom-up

text: predictions

text: vˆP

text: Recalling

text: Eq.

text: we

text: obtain

text: a

text: recurrence

text: relation

text: O

text: e

text: λ

text: D

text: W

text: e

text: λ2

text: W

text: e

text: k

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: to which the weights deviate from their optimal setting, we introduce the convex loss function (cid:8) (cid:9) L 1 ∗ − ∗ − IP = Tr (WIP WIP )T (WIP WIP ) , (S25) k 2 k,k k,k k,k k,k ∗ where Tr(M) denotes the trace of matrix M and WIP = gB+glk WPP , as deﬁned in Eq. S2. k,k g +g +g k+1,k B A lk Starting from the pyramidal-to-interneuron synaptic plasticity rule (8), we express the interneu- ron somatic potential in convex combination form (S3) and then expand to ﬁrst order around vˆI , k (cid:0) (cid:1) (cid:0) − (cid:1) ∆WIP = ηIP φ(uI ) φ(vˆI ) (rP)T k,k k,k k k k − (cid:0) O(cid:1) = ηIP λ DIP uP vˆI (rP)T + (λ2) k,k k k+1 k k g ∗ − O O = ηIP λ B DIP WIP WIP Q + (λ2) + (λ α). (S26) k,k g + g k k,k k,k k lk B Matrix Q = rP (rP)T denotes the outer product, and DIP is a diagonal matrix with i-th diagonal k k k k (cid:48) entry equal to φ (vˆI ). k,i For simplicity, we ignore ﬂuctuations arising from the stochastic sequential presentation of pat- terns (Bottou, 1998) and look only at the expected synaptic dynamics1. We absorb irrelevant scale − factors and to avoid a vanishing update we rescale the learning rate ηˆIP by λ 1. Then, taking the k,k → → limit λ 0, α 0 as in the previous sections yields (cid:2) (cid:3) (cid:0) (cid:1) (cid:2) (cid:3) ∗ − (cid:2) (cid:3) E ∆WIP = ηˆIP WIP WIP E DIPQ k,k k,k k,k k,k k k L − ∂ IP = ηˆIP k E DIPQ . (S27) k,k ∂WIP k k k,k The expectation is taken over the pattern ensemble. In the last equality above, we used the fact L ∗− that the gradient of IP with respect to the lateral weights WIP is given by the difference WIP WIP . k k,k k,k k,k As long as the expectation on the right-hand side of (S27) is positive deﬁnite, the synaptic dy- L namics is within 90o of the gradient and thus leads to the unique minimum of IP. This condition is k 1This can be understood as a batch learning protocol, where weight changes are accumulated in the limit of many patterns before being effectively consolidated as a synaptic update. 37 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: to

text: which

text: the

text: weights

text: deviate

text: from

text: their

text: optimal

text: setting,

text: we

text: introduce

text: the

text: convex

text: loss

text: function

text: L

text: IP

text: Tr

text: (WIP

text: WIP

text: )T

text: (WIP

text: WIP

text: k

text: k,k

text: k,k

text: k,k

text: k,k

text: where

text: Tr(M)

text: denotes

text: the

text: trace

text: of

text: matrix

text: M

text: and

text: WIP

text: gB+glk

text: WPP

text: as

text: deﬁned

text: in

text: Eq.

text: k,k

text: g

text: +g

text: +g

text: B

text: A

text: lk

text: Starting

text: from

text: the

text: pyramidal-to-interneuron

text: synaptic

text: plasticity

text: rule

text: we

text: express

text: the

text: interneu-

text: ron

text: somatic

text: potential

text: in

text: convex

text: combination

text: form

text: and

text: then

text: expand

text: to

text: ﬁrst

text: order

text: around

text: vˆI

text: k

text: ∆WIP

text: ηIP

text: φ(uI

text: φ(vˆI

text: (rP)T

text: k,k

text: k,k

text: k

text: k

text: k

text: ηIP

text: λ

text: DIP

text: uP

text: vˆI

text: (rP)T

text: k,k

text: k

text: k

text: k

text: g

text: O

text: O

text: ηIP

text: λ

text: B

text: DIP

text: WIP

text: WIP

text: Q

text: (λ

text: α).

text: k,k

text: g

text: g

text: k

text: k,k

text: k,k

text: k

text: lk

text: B

text: Matrix

text: Q

text: rP

text: (rP)T

text: denotes

text: the

text: outer

text: product,

text: and

text: DIP

text: is

text: a

text: diagonal

text: matrix

text: with

text: i-th

text: diagonal

text: k

text: k

text: k

text: k

text: entry

text: equal

text: to

text: φ

text: (vˆI

text: k,i

text: For

text: simplicity,

text: we

text: ignore

text: ﬂuctuations

text: arising

text: from

text: the

text: stochastic

text: sequential

text: presentation

text: of

text: pat-

text: terns

text: (Bottou,

text: and

text: look

text: only

text: at

text: the

text: expected

text: synaptic

text: We

text: absorb

text: irrelevant

text: scale

text: factors

text: and

text: to

text: avoid

text: a

text: vanishing

text: update

text: we

text: rescale

text: the

text: learning

text: rate

text: ηˆIP

text: by

text: λ

text: Then,

text: taking

text: the

text: k,k

text: limit

text: λ

text: α

text: as

text: in

text: the

text: previous

text: sections

text: yields

text: E

text: ∆WIP

text: ηˆIP

text: WIP

text: WIP

text: E

text: DIPQ

text: k,k

text: k,k

text: k,k

text: k,k

text: k

text: k

text: L

text: IP

text: ηˆIP

text: k

text: E

text: DIPQ

text: k,k

text: ∂WIP

text: k

text: k

text: k,k

text: The

text: expectation

text: is

text: taken

text: over

text: the

text: pattern

text: ensemble.

text: In

text: the

text: last

text: equality

text: above,

text: we

text: used

text: the

text: fact

text: L

text: that

text: the

text: gradient

text: of

text: IP

text: with

text: respect

text: to

text: the

text: lateral

text: weights

text: WIP

text: is

text: given

text: by

text: the

text: difference

text: WIP

text: WIP

text: k

text: k,k

text: k,k

text: k,k

text: As

text: long

text: as

text: the

text: expectation

text: on

text: the

text: right-hand

text: side

text: of

text: is

text: positive

text: deﬁnite,

text: the

text: synaptic

text: dy-

text: L

text: namics

text: is

text: within

text: of

text: the

text: gradient

text: and

text: thus

text: leads

text: to

text: the

text: unique

text: minimum

text: of

text: IP.

text: This

text: condition

text: is

text: k

text: can

text: be

text: understood

text: as

text: a

text: batch

text: learning

text: protocol,

text: where

text: weight

text: changes

text: are

text: accumulated

text: in

text: the

text: limit

text: of

text: many

text: patterns

text: before

text: being

text: effectively

text: consolidated

text: as

text: a

text: synaptic

text: update.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: easily met in practice. For linear neurons, it amounts to requiring that the correlation matrix E[Q] is R positive deﬁnite. In other words, the patterns have to span Nk , with Nk being the number of pyra- midal neurons at area k. This is fulﬁlled when uncorrelated background noise currents are present, and it is likely the case for deterministic networks solving nontrivial tasks. For nonlinear neurons with a monotonically increasing transfer function φ, saturation can lead to a matrix numerically close to singular and slow down learning. A weight matrix initialization that sets the neurons operating far from saturation is therefore an appropriate choice. A mathematical analysis of the coupled system deﬁned by the various plasticity rules acting in concert is rather involved and beyond the scope of this note. However, the learning of apical- targetting interneuron-to-pyramidal synapses can be studied in isolation by invoking a separation of timescales argument. To proceed, we assume that pyramidal-to-interneuron synapses are ideally ∗ set, WIP = WIP . In practice, this translates to a choice of a small effective learning rate for apical- k,k k,k targetting weights ηPI . Note that, indirectly, this requirement also imposes a constraint on how fast k,k top-down pyramidal-to-pyramidal weights WPP can evolve. This is the parameter regime explored k,k+1 in the main text simulations with plastic top-down weights, Fig. 5. We can then proceed in a similar manner to the previous analysis, as we brieﬂy outline below. ∗ − Recalling from Eq. S1 that WPI = WPP , we deﬁne the loss function k,k k,k+1 (cid:8) (cid:9) L 1 ∗ − ∗ − PI = Tr (WPI WPI )T (WPI WPI ) . (S28) k 2 k,k k,k k,k k,k → After some manipulation, as λ 0 the expected synaptic change can be written as (cid:2) (cid:3) (cid:2) (cid:3) L − ∂ PI E ∆WPI = ηPI k E rP (rP )T , (S29) k,k k,k ∂WPI k+1 k+1 k,k which leads us to conclude that the weights converge to the appropriate values, provided that the correlation matrix of area k +1 activity patterns is positive deﬁnite. 38 ##LTLine## ##LTLine##

text: easily

text: met

text: in

text: practice.

text: For

text: linear

text: neurons,

text: it

text: amounts

text: to

text: requiring

text: that

text: the

text: correlation

text: matrix

text: E[Q]

text: is

text: R

text: positive

text: deﬁnite.

text: In

text: other

text: words,

text: the

text: patterns

text: have

text: to

text: span

text: Nk

text: with

text: Nk

text: being

text: the

text: number

text: of

text: pyra-

text: midal

text: neurons

text: at

text: area

text: k.

text: This

text: is

text: fulﬁlled

text: when

text: uncorrelated

text: background

text: noise

text: currents

text: are

text: present,

text: and

text: it

text: is

text: likely

text: the

text: case

text: for

text: deterministic

text: networks

text: solving

text: nontrivial

text: tasks.

text: For

text: nonlinear

text: neurons

text: with

text: a

text: monotonically

text: increasing

text: transfer

text: function

text: φ,

text: saturation

text: can

text: lead

text: to

text: a

text: matrix

text: numerically

text: close

text: to

text: singular

text: and

text: slow

text: down

text: learning.

text: A

text: weight

text: matrix

text: initialization

text: that

text: sets

text: the

text: neurons

text: operating

text: far

text: from

text: saturation

text: is

text: therefore

text: an

text: appropriate

text: choice.

text: A

text: mathematical

text: analysis

text: of

text: the

text: coupled

text: system

text: deﬁned

text: by

text: the

text: various

text: plasticity

text: rules

text: acting

text: in

text: concert

text: is

text: rather

text: involved

text: and

text: beyond

text: the

text: scope

text: of

text: this

text: note.

text: However,

text: the

text: learning

text: of

text: apical-

text: targetting

text: interneuron-to-pyramidal

text: synapses

text: can

text: be

text: studied

text: in

text: isolation

text: by

text: invoking

text: a

text: separation

text: of

text: timescales

text: argument.

text: To

text: proceed,

text: we

text: assume

text: that

text: pyramidal-to-interneuron

text: synapses

text: are

text: ideally

text: set,

text: WIP

text: WIP

text: In

text: practice,

text: this

text: translates

text: to

text: a

text: choice

text: of

text: a

text: small

text: effective

text: learning

text: rate

text: for

text: apical-

text: k,k

text: k,k

text: targetting

text: weights

text: ηPI

text: Note

text: that,

text: indirectly,

text: this

text: requirement

text: also

text: imposes

text: a

text: constraint

text: on

text: how

text: fast

text: k,k

text: top-down

text: pyramidal-to-pyramidal

text: weights

text: WPP

text: can

text: evolve.

text: This

text: is

text: the

text: parameter

text: regime

text: explored

text: in

text: the

text: main

text: text

text: simulations

text: with

text: plastic

text: top-down

text: weights,

text: Fig.

text: We

text: can

text: then

text: proceed

text: in

text: a

text: similar

text: manner

text: to

text: the

text: previous

text: analysis,

text: as

text: we

text: brieﬂy

text: outline

text: below.

text: Recalling

text: from

text: Eq.

text: that

text: WPI

text: WPP

text: we

text: deﬁne

text: the

text: loss

text: function

text: k,k

text: L

text: PI

text: Tr

text: (WPI

text: WPI

text: )T

text: (WPI

text: WPI

text: k

text: k,k

text: k,k

text: k,k

text: k,k

text: After

text: some

text: manipulation,

text: as

text: λ

text: the

text: expected

text: synaptic

text: change

text: can

text: be

text: written

text: as

text: L

text: PI

text: E

text: ∆WPI

text: ηPI

text: k

text: E

text: rP

text: (rP

text: )T

text: k,k

text: k,k

text: ∂WPI

text: k,k

text: which

text: leads

text: us

text: to

text: conclude

text: that

text: the

text: weights

text: converge

text: to

text: the

text: appropriate

text: values,

text: provided

text: that

text: the

text: correlation

text: matrix

text: of

text: area

text: k

text: activity

text: patterns

text: is

text: positive

text: deﬁnite.

text: ##LTLine##

text: ##LTLine##


text: We let learning occur in continuous time without pauses or alternations in plasticity as input patterns are sequentially presented. This is in contrast to previous learning models that rely on computing activity differences over distinct phases, requiring temporally nonlocal computation, or globally coordinated plasticity rule switches (Hinton and McClelland, 1988; O’Reilly, 1996; Xie and Seung, 2003; Scellier and Bengio, 2017; Guerguiev et al., 2017). Furthermore, we relaxed the (cid:0) (cid:1) bottom-up vs. top-down weight symmetry imposed by the backprop algorithm and kept the top-down ∼ weights WPP ﬁxed. Feedback WPP weights quickly aligned to 45o of the forward weights WPP T , 1,2 1,2 2,1 in line with the recently discovered feedback alignment phenomenon (Lillicrap et al., 2016). This simpliﬁes the architecture, because top-down and interneuron-to-pyramidal synapses need not to be changed. Finally, to test the robustness of the network, we injected a weak noise current to every neuron, as a simple model for uncorrelated background activity (see Methods). Our network was still able to learn this harder task (Fig. 3B), performing considerably better than a shallow learner where only output weights were adjusted (Fig. 3C). Useful changes were thus made to hidden area 1 bottom-up weights; the network effectively solved the credit assignment problem. Multi-area network learns to discriminate handwritten digits Next, we turn to a standard machine learning problem, the classiﬁcation of handwritten digits from the MNIST database. This data set is popularly used to study the performance of learning models, including various artiﬁcial neural networks trained with backprop. Notably, shallow models (e.g., logistic regression) or networks trained with plain Hebbian learning alone suffer from poor performance and do not offer a remedy for the problem. We wondered how our model would fare in this real-world benchmark, in particular whether the prediction errors computed by the interneuron microcircuit would allow learning the weights of a hierarchical nonlinear network with multiple hidden areas. To that end, we trained a deeper, larger 4-area network (with 784-500-500-10 pyramidal neurons, Fig. 4A) by pairing digit images with teaching inputs that nudged the 10 output neurons towards the correct class pattern. To speed up the experiments we studied a simpliﬁed network dynamics which determined compartmental potentials without requiring a full neuronal relaxation procedure (see Methods). As in the previous experiments, synaptic weights were randomly initialized and set to a self-predicting conﬁguration where interneurons cancelled top-down inputs, rendering the apical compartments silent before training started. Top-down and interneuron-to-pyramidal weights were kept ﬁxed. The network was able to achieve a test error of 1.96%, Fig. 4B, a ﬁgure not overly far from the 9

text: We

text: let

text: learning

text: occur

text: in

text: continuous

text: time

text: without

text: pauses

text: or

text: alternations

text: in

text: plasticity

text: as

text: input

text: patterns

text: are

text: sequentially

text: presented.

text: This

text: is

text: in

text: contrast

text: to

text: previous

text: learning

text: models

text: that

text: rely

text: on

text: computing

text: activity

text: differences

text: over

text: distinct

text: phases,

text: requiring

text: temporally

text: nonlocal

text: computation,

text: or

text: globally

text: coordinated

text: plasticity

text: rule

text: switches

text: (Hinton

text: and

text: McClelland,

text: O’Reilly,

text: Xie

text: and

text: Seung,

text: Scellier

text: and

text: Bengio,

text: Guerguiev

text: et

text: al.,

text: Furthermore,

text: we

text: relaxed

text: the

text: bottom-up

text: vs.

text: top-down

text: weight

text: symmetry

text: imposed

text: by

text: the

text: backprop

text: algorithm

text: and

text: kept

text: the

text: top-down

text: weights

text: WPP

text: ﬁxed.

text: Feedback

text: WPP

text: weights

text: quickly

text: aligned

text: to

text: of

text: the

text: forward

text: weights

text: WPP

text: T

text: in

text: line

text: with

text: the

text: recently

text: discovered

text: feedback

text: alignment

text: phenomenon

text: (Lillicrap

text: et

text: al.,

text: This

text: simpliﬁes

text: the

text: architecture,

text: because

text: top-down

text: and

text: interneuron-to-pyramidal

text: synapses

text: need

text: not

text: to

text: be

text: changed.

text: Finally,

text: to

text: test

text: the

text: robustness

text: of

text: the

text: network,

text: we

text: injected

text: a

text: weak

text: noise

text: current

text: to

text: every

text: neuron,

text: as

text: a

text: simple

text: model

text: for

text: uncorrelated

text: background

text: activity

text: (see

text: Methods).

text: Our

text: network

text: was

text: still

text: able

text: to

text: learn

text: this

text: harder

text: task

text: (Fig.

text: performing

text: considerably

text: better

text: than

text: a

text: shallow

text: learner

text: where

text: only

text: output

text: weights

text: were

text: adjusted

text: (Fig.

text: Useful

text: changes

text: were

text: thus

text: made

text: to

text: hidden

text: area

text: bottom-up

text: weights;

text: the

text: network

text: effectively

text: solved

text: the

text: credit

text: assignment

text: problem.

text: Multi-area

text: network

text: learns

text: to

text: discriminate

text: handwritten

text: digits

text: Next,

text: we

text: turn

text: to

text: a

text: standard

text: machine

text: learning

text: problem,

text: the

text: classiﬁcation

text: of

text: handwritten

text: digits

text: from

text: the

text: MNIST

text: database.

text: This

text: data

text: set

text: is

text: popularly

text: used

text: to

text: study

text: the

text: performance

text: of

text: learning

text: models,

text: including

text: various

text: artiﬁcial

text: neural

text: networks

text: trained

text: with

text: backprop.

text: Notably,

text: shallow

text: models

text: (e.g.,

text: logistic

text: regression)

text: or

text: networks

text: trained

text: with

text: plain

text: Hebbian

text: learning

text: alone

text: suffer

text: from

text: poor

text: performance

text: and

text: do

text: not

text: offer

text: a

text: remedy

text: for

text: the

text: problem.

text: We

text: wondered

text: how

text: our

text: model

text: would

text: fare

text: in

text: this

text: real-world

text: benchmark,

text: in

text: particular

text: whether

text: the

text: prediction

text: errors

text: computed

text: by

text: the

text: interneuron

text: microcircuit

text: would

text: allow

text: learning

text: the

text: weights

text: of

text: a

text: hierarchical

text: nonlinear

text: network

text: with

text: multiple

text: hidden

text: areas.

text: To

text: that

text: end,

text: we

text: trained

text: a

text: deeper,

text: larger

text: network

text: (with

text: pyramidal

text: neurons,

text: Fig.

text: by

text: pairing

text: digit

text: images

text: with

text: teaching

text: inputs

text: that

text: nudged

text: the

text: output

text: neurons

text: towards

text: the

text: correct

text: class

text: pattern.

text: To

text: speed

text: up

text: the

text: experiments

text: we

text: studied

text: a

text: simpliﬁed

text: network

text: dynamics

text: which

text: determined

text: compartmental

text: potentials

text: without

text: requiring

text: a

text: full

text: neuronal

text: relaxation

text: procedure

text: (see

text: Methods).

text: As

text: in

text: the

text: previous

text: experiments,

text: synaptic

text: weights

text: were

text: randomly

text: initialized

text: and

text: set

text: to

text: a

text: self-predicting

text: conﬁguration

text: where

text: interneurons

text: cancelled

text: top-down

text: inputs,

text: rendering

text: the

text: apical

text: compartments

text: silent

text: before

text: training

text: started.

text: Top-down

text: and

text: interneuron-to-pyramidal

text: weights

text: were

text: kept

text: ﬁxed.

text: The

text: network

text: was

text: able

text: to

text: achieve

text: a

text: test

text: error

text: of

text: Fig.

text: a

text: ﬁgure

text: not

text: overly

text: far

text: from

text: the


text: 7.1 Stability Boundary Computation For simulation purposes, we consider the swing dynamics of N +1 generator sys- tem as the model, δ˙i = ωi, ω˙i = Pi −Diωi −N∑+1EiEjYijsin(δi −δj), i = 1,...N. (12) j=1 The (N +1)th generator is taken as the reference, by making EN+1 = 1, and δN+1 = 0. Let us deﬁne the state variable for the dynamical system as, x := [ δ1, ..., δN, ω1, ,..., ωN]T ∈ Rn. This system corresponds to a speciﬁc set of values of the P’s, and also the system is at i a stable equilibrium of the (12). Let, a fault occurs, at tf , and is cleared at t = 0. The source of the fault may be change in the electric power input at generator i, which is given by P, or short circuit, given by changes in the values Y . Once the fault is cleared i ij the system dynamics is changed according to the new parameters of the system. The phase portrait of two generator system is depicted in Fig. 5. The stable manifold of the type-1 saddle point forms the stability boundary. The stable equilibrium point is shown as a black circle and the type -1 saddle point is shown as a red cross. Figure 6 shows the plot of the normal repulsion rate over a time window of 5 sec, which is typical time interval for short term transient stability. Over this time window, it can be observed that a relatively larger region around the stable manifold of the type-1 saddle point show normal repulsion more than 1. The region, which has more than 1 repulsion rate, indicates the ﬁnite time stable region. This can not be captured by the asymptotic approaches. It can be observed from Fig. 7, as the time window is extended to 35 sec, the stable manifolds of the type-1 saddle point emerges as the material surface with positive repulsion rate, which matches with the asymptotic analysis. It demonstrates, our approach gives a concrete way to provide the region of stability over a ﬁnite time window. Figures 8, and 9 show the plot of FTLE over time intervals of 5, and 35 sec. It can be observed that the FTLE can locally detect the saddle point. FTLE can not detect globally the stability boundaries, as it becomes negative along the stable manifold of the saddle point. Whereas, near the saddle point, the unstable manifold has locally positive FTLE. This is because the unstable manifold of the saddle is locally tangentially expanding. Also, we have observed the ridges in the normal repulsion rate or FTLE ﬁeld preserve themselves under small perturbations or noise. Next, we demonstrate with simulations on 39 bus system, that the normal expansion rate and LE can be used for online stability monitoring. 7.2 Online Transient Stability Monitoring for New England 39 Bus System In this subsection, we show application of normal expansion rate ρ(t) , and LE λ(t) as stability certiﬁcates in online monitoring. Also, we compute stability margins 17

label: section

text: Stability

text: Boundary

text: Computation

text: For

text: simulation

text: purposes,

text: we

text: consider

text: the

text: swing

text: dynamics

text: of

text: N

text: generator

text: sys-

text: tem

text: as

text: the

text: model,

text: δ˙i

text: ωi,

text: ω˙i

text: Pi

text: −Diωi

text: −δj),

text: i

text: The

text: (N

text: generator

text: is

text: taken

text: as

text: the

text: reference,

text: by

text: making

text: and

text: Let

text: us

text: deﬁne

text: the

text: state

text: variable

text: for

text: the

text: dynamical

text: system

text: as,

text: x

text: δN,

text: ωN]T

text: Rn.

text: This

text: system

text: corresponds

text: to

text: a

text: speciﬁc

text: set

text: of

text: values

text: of

text: the

text: P’s,

text: and

text: also

text: the

text: system

text: is

text: at

text: i

text: a

text: stable

text: equilibrium

text: of

text: the

text: Let,

text: a

text: fault

text: occurs,

text: at

text: tf

text: and

text: is

text: cleared

text: at

text: t

text: The

text: source

text: of

text: the

text: fault

text: may

text: be

text: change

text: in

text: the

text: electric

text: power

text: input

text: at

text: generator

text: i,

text: which

text: is

text: given

text: by

text: P,

text: or

text: short

text: circuit,

text: given

text: by

text: changes

text: in

text: the

text: values

text: Y

text: Once

text: the

text: fault

text: is

text: cleared

text: i

text: ij

text: the

text: system

text: dynamics

text: is

text: changed

text: according

text: to

text: the

text: new

text: parameters

text: of

text: the

text: system.

text: The

text: phase

text: portrait

text: of

text: two

text: generator

text: system

text: is

text: depicted

text: in

text: Fig.

text: The

text: stable

text: manifold

text: of

text: the

text: saddle

text: point

text: forms

text: the

text: stability

text: boundary.

text: The

text: stable

text: equilibrium

text: point

text: is

text: shown

text: as

text: a

text: black

text: circle

text: and

text: the

text: type

text: saddle

text: point

text: is

text: shown

text: as

text: a

text: red

text: cross.

text: Figure

text: shows

text: the

text: plot

text: of

text: the

text: normal

text: repulsion

text: rate

text: over

text: a

text: time

text: window

text: of

text: sec,

text: which

text: is

text: typical

text: time

text: interval

text: for

text: short

text: term

text: transient

text: stability.

text: Over

text: this

text: time

text: window,

text: it

text: can

text: be

text: observed

text: that

text: a

text: relatively

text: larger

text: region

text: around

text: the

text: stable

text: manifold

text: of

text: the

text: saddle

text: point

text: show

text: normal

text: repulsion

text: more

text: than

text: The

text: region,

text: which

text: has

text: more

text: than

text: repulsion

text: rate,

text: indicates

text: the

text: ﬁnite

text: time

text: stable

text: region.

text: This

text: can

text: not

text: be

text: captured

text: by

text: the

text: asymptotic

text: approaches.

text: It

text: can

text: be

text: observed

text: from

text: Fig.

text: as

text: the

text: time

text: window

text: is

text: extended

text: to

text: sec,

text: the

text: stable

text: manifolds

text: of

text: the

text: saddle

text: point

text: emerges

text: as

text: the

text: material

text: surface

text: with

text: positive

text: repulsion

text: rate,

text: which

text: matches

text: with

text: the

text: asymptotic

text: analysis.

text: It

text: demonstrates,

text: our

text: approach

text: gives

text: a

text: concrete

text: way

text: to

text: provide

text: the

text: region

text: of

text: stability

text: over

text: a

text: ﬁnite

text: time

text: window.

text: Figures

text: and

text: show

text: the

text: plot

text: of

text: FTLE

text: over

text: time

text: intervals

text: of

text: and

text: sec.

text: It

text: can

text: be

text: observed

text: that

text: the

text: FTLE

text: can

text: locally

text: detect

text: the

text: saddle

text: point.

text: FTLE

text: can

text: not

text: detect

text: globally

text: the

text: stability

text: boundaries,

text: as

text: it

text: becomes

text: negative

text: along

text: the

text: stable

text: manifold

text: of

text: the

text: saddle

text: point.

text: Whereas,

text: near

text: the

text: saddle

text: point,

text: the

text: unstable

text: manifold

text: has

text: locally

text: positive

text: FTLE.

text: This

text: is

text: because

text: the

text: unstable

text: manifold

text: of

text: the

text: saddle

text: is

text: locally

text: tangentially

text: expanding.

text: Also,

text: we

text: have

text: observed

text: the

text: ridges

text: in

text: the

text: normal

text: repulsion

text: rate

text: or

text: FTLE

text: ﬁeld

text: preserve

text: themselves

text: under

text: small

text: perturbations

text: or

text: noise.

text: Next,

text: we

text: demonstrate

text: with

text: simulations

text: on

text: bus

text: system,

text: that

text: the

text: normal

text: expansion

text: rate

text: and

text: LE

text: can

text: be

text: used

text: for

text: online

text: stability

text: monitoring.

text: Online

text: Transient

text: Stability

text: Monitoring

text: for

text: New

text: England

text: Bus

text: System

text: In

text: this

text: subsection,

text: we

text: show

text: application

text: of

text: normal

text: expansion

text: rate

text: ρ(t)

text: and

text: LE

text: λ(t)

text: as

text: stability

text: certiﬁcates

text: in

text: online

text: monitoring.

text: Also,

text: we

text: compute

text: stability

text: margins


text: vector η0(x0), we have, (cid:0) (cid:1) 1 ∑ni=1 Λi(x0,n01,ε0,−T) η0(cid:48)ui(x0,n0,ε0,−T) 2 = 1 (cid:48) = ∑ni=1 Λi(x0,n01,ε0,−T)η0η0 − 1 Λn K ( T,ε0) > > . ∑N 1 − n n i=1 Λi(x0,n0,ε0, T) − The ε0 can be made sufﬁciently small such that, K ( T,ε0) > n. This will give, (cid:0) (cid:1) 1 (cid:16) ∑ni=1 Λi(x0,n01,ε0,−T) η0(cid:48)ui(x0,n0,ε0,(cid:17)−T) 2 > 1, (21) 1 − > 1, (22) − − − − 1 η0T (∇Φ(x0 ε0n0, T)))T ∇Φ(x0 ε0n0, T) η0 − ρ(x0 ε0n0,T) > 1. (23) Similarly, we use, − − ∂Φ − e( T,ε0) = 2ε ∂x (x0 +ε0n0(x0), T)n0(x0) − − and use the successive steps to prove ρ(x0 ε0n0, T) > 1 . Now let us deﬁne a set, { | (cid:107) − (cid:107)≤ } Nε(xs) := x x xs ε , − where ε < ε0. This gives us, ρ(x, T) > 1 for all x, which is at most ε away from the point xs. Hence the proof. Below, we provide the proof of Proposition 23. Proof. Let us deﬁne − ζ(t) := Φ(x0,t +∆t) Φ(x0,t), ∈ \ for initial (cid:54)condition x∈0 A(xs\) xs, and ∆t is a small positive number. ∈It can be\observed that ζ(0) = 0, as x0 A(xs) xs. Also, it can be observed Φ(x0,∆t) A(xs) xs. Con- sidering two initial conditions x0, and Φ(x0,∆t) the following assertions can be made - tshuecrhetehxaitst(cid:107)sΦa (Tx1∗0,stu+ch∆tth)a−t (cid:107)xΦs (cid:107)(<x0,(cid:107)tζ)(2−0)(cid:107)xsf(cid:107)o<r al(cid:107)lζt(20≥)(cid:107)Tf2o∗r. aWllet d≥eﬁT1n∗i.nAglTso∗,:t=hemreaexx(iTs1t∗s,aT2T∗)2∗. Using triangle inequality (cid:107) − (cid:107) Φ(x0,t +∆t) Φ(x0,t) (cid:107) − (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) ∀ ≥ ∗ < Φ(x0,t +∆t) xs + xs Φ(x0,t) < ζ0 , t T . (cid:107) − (cid:107) (cid:107) (cid:107) ≥ ∗ Hence, we can assert that, Φ(≥x0,t∗+ ∆t) Φ(x0,t) < ζ(0) for all t T . This gives us, λ(x0,0,t) < 0 for all t T . 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: vector

text: we

text: have,

text: Λn

font: ZADGCU+StandardSymL

text: K

text: ∑N

text: n

text: n

text: T)

text: The

text: ε0

text: can

text: be

text: made

text: sufﬁciently

text: small

text: such

text: that,

text: K

text: n.

text: This

text: will

text: give,

text: T)))T

text: T)

text: η0

text: Similarly,

text: we

text: use,

text: ∂Φ

text: e(

text: 2ε

text: ∂x

text: and

text: use

text: the

text: successive

text: steps

text: to

text: prove

text: T)

text: Now

text: let

text: us

text: deﬁne

text: a

text: set,

text: Nε(xs)

text: x

text: x

text: xs

text: ε

text: where

text: ε

text: This

text: gives

text: us,

text: ρ(x,

text: T)

text: for

text: all

text: x,

text: which

text: is

text: at

text: most

text: ε

text: away

text: from

text: the

text: point

text: xs.

text: Hence

text: the

text: proof.

text: Below,

text: we

text: provide

text: the

text: proof

text: of

text: Proposition

text: Proof.

text: Let

text: us

text: deﬁne

text: ζ(t)

text: +∆t)

text: for

text: initial

text: A(xs\)

text: xs,

text: and

text: ∆t

text: is

text: a

text: small

text: positive

text: number.

text: ∈It

text: can

text: be\observed

text: that

text: as

text: A(xs)

text: xs.

text: Also,

text: it

text: can

text: be

text: observed

text: A(xs)

text: xs.

text: Con-

text: sidering

text: two

text: initial

text: conditions

text: and

text: the

text: following

text: assertions

text: can

text: be

text: made

text: aWllet

text: Using

text: triangle

text: inequality

text: +∆t)

text: +∆t)

text: xs

text: xs

text: ζ0

text: t

text: T

text: Hence,

text: we

text: can

text: assert

text: that,

text: ∆t)

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text: gives

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text: [14] D. Novosel, V. Madani, B. Bhargava, K. Vu, and J. Cole. Dawn of the grid synchronization: Beneﬁts, practical applications, and deployment strategies for wide area monitoring, protection, and control. IEEE Power Energy Magazine, 6:49–60, 2008. [15] A. G. Phadke. Synchronized phasor measurements in power systems. IEEE Computer Applications in Power, 6:10–15, 1993. [16] A. G. Phadke and J. S. Thorp. Synchronized Phasor Measurements and their Applications. Springer, New York, 2008. [17] A. G. Phadke, H. Voslkis, R. M. de Morales, T. Bi, R. N. Nayak, Y. K. Sehgal, S. Sen, W. Sattinger, E. Martinez, O. Samuelsson, D. Novosel, V. Madani, and Y. A. Kulikov. The wide world of wide-area measurements. IEEE Power Energy Magazine, 6:52–65, 2008. [18] Shawn C. Shadden, Francois Lekien, and Jerrold E. Marsden. Deﬁnition and properties of lagrangian coherent structures from ﬁnite-time lyapunov expo- nents in two-dimensional aperiodiclows. Physica D: Nonlinear Phenomena, 212(3):271 – 304, 2005. [19] Pravin Varaiya, Felix F Wu, and Rong-Liang Chen. Direct methods for transient stability analysis of power systems: Recent results. Proceedings of the IEEE, 73(12):1703–1715, 1985. [20] Stephen Wiggins. Normally hyperbolic invariant manifolds in dynamical systems. Number 105. Springer, 1994. [21] Jie Yan, Chen-Ching Liu, and U. Vaidya. PMU-based monitoring of rotor angle dynamics. IEEE Transactions on Power Systems, 26(4):2125–2133, 2011. [22] R.D. Zimmerman, C.E. Murillo-Snchez, and R.J.Thomas. Matpower: Steady- state operations, planning, and analysis tools for power systems research and ed- ucation. IEEE Transactions on Power Systems, 26(1):12–19, 2011. 29

label: reference

text: D.

text: Novosel,

text: V.

text: Madani,

text: B.

text: Bhargava,

text: K.

text: Vu,

text: and

text: J.

text: Cole.

text: Dawn

text: of

text: the

text: grid

text: synchronization:

text: Beneﬁts,

text: practical

text: applications,

text: and

text: deployment

text: strategies

text: for

text: wide

text: area

text: monitoring,

text: protection,

text: and

text: control.

text: IEEE

text: Power

text: Energy

text: Magazine,

text: A.

text: G.

text: Phadke.

text: Synchronized

text: phasor

text: measurements

text: in

text: power

text: systems.

text: IEEE

text: Computer

text: Applications

text: in

text: Power,

text: A.

text: G.

text: Phadke

text: and

text: J.

text: S.

text: Thorp.

text: Synchronized

text: Phasor

text: Measurements

text: and

text: their

text: Applications.

text: Springer,

text: New

text: York,

text: A.

text: G.

text: Phadke,

text: H.

text: Voslkis,

text: R.

text: M.

text: de

text: Morales,

text: T.

text: Bi,

text: R.

text: N.

text: Nayak,

text: Y.

text: K.

text: Sehgal,

text: S.

text: Sen,

text: W.

text: Sattinger,

text: E.

text: Martinez,

text: O.

text: Samuelsson,

text: D.

text: Novosel,

text: V.

text: Madani,

text: and

text: Y.

text: A.

text: Kulikov.

text: The

text: wide

text: world

text: of

text: wide-area

text: measurements.

text: IEEE

text: Power

text: Energy

text: Magazine,

text: Shawn

text: C.

text: Shadden,

text: Francois

text: Lekien,

text: and

text: Jerrold

text: E.

text: Marsden.

text: Deﬁnition

text: and

text: properties

text: of

text: lagrangian

text: coherent

text: structures

text: from

text: ﬁnite-time

text: lyapunov

text: expo-

text: nents

text: in

text: two-dimensional

text: aperiodiclows.

text: Physica

text: D:

text: Nonlinear

text: Phenomena,

text: Pravin

text: Varaiya,

text: Felix

text: F

text: Wu,

text: and

text: Rong-Liang

text: Chen.

text: Direct

text: methods

text: for

text: transient

text: stability

text: analysis

text: of

text: power

text: systems:

text: Recent

text: results.

text: Proceedings

text: of

text: the

text: IEEE,

text: Stephen

text: Wiggins.

text: Normally

text: hyperbolic

text: invariant

text: manifolds

text: in

text: dynamical

text: systems.

text: Number

text: Springer,

text: Jie

text: Yan,

text: Chen-Ching

text: Liu,

text: and

text: U.

text: Vaidya.

text: PMU-based

text: monitoring

text: of

text: rotor

text: angle

text: dynamics.

text: IEEE

text: Transactions

text: on

text: Power

text: Systems,

text: R.D.

text: Zimmerman,

text: C.E.

text: Murillo-Snchez,

text: and

text: R.J.Thomas.

text: Matpower:

text: Steady-

text: state

text: operations,

text: planning,

text: and

text: analysis

text: tools

text: for

text: power

text: systems

text: research

text: and

text: ed-

text: ucation.

text: IEEE

text: Transactions

text: on

text: Power

text: Systems,


text: If v(x0,n0,t) > 1, then the normal growth along M (t) dominates the largest tangen- tial growth rate along M (t). From the point of view of computation, both the normal rate and ratio can be computed using the following formulas [9]. (cid:112) 1 ρ(x0,n0,t) = (cid:104) − (cid:105), (7) n0,[Ct(x0)] 1n0 ∗ where, Ct(x0) := ∇Φ(x0,t) ∇Φ(x0,t). (cid:112) v(x0,n0,t) = |e0|=1,em0∈iTnx0M(0) (cid:104)n0ρ,([xC0t,(nx00,)t]−) 1n0(cid:105). (8) With the aid of normal repulsion rate and ratio, we deﬁne ﬁnite-time repelling (at- tracting) material surfaces. Deﬁnition 13 (Normally repelling material surface[9]). A material surface M (t) is normally repelling over time interval [0,T] if there exist numbers α,β > 0, such that, ∈ ∈ for all points x0 M (0), and unit normal vector n0 Nx0M (0), we have [9], ≥ ρ(x0,n0,t) eαt, ≥ v(x0,n0,t) eβt. A material surface M (t) is normally attracting over time interval [0,T] if it is normally repelling over [0,T] in backward time. We call M (t) hyperbolic over [0,T] if it is normally attracting or normally repelling over [0,T]. All repelling or attracting material surfaces are not equally repelling or attracting. Some material surfaces are more repelling or attracting than others. Material surfaces that are maximally attracting or repelling occupy special place and name in the theory of normally hyperbolic invariant manifolds. They are called as Lagrangian coherent structures. We have following deﬁnitions in this direction. Deﬁnition 14 (Lagrangian Coherent Surface [9]). Assume that M (t) is a normally ∈ repelling material surface over t [0,T]. We call M (t) a repelling Lagrangian Co- herent Surface (LCS) over [0,T] if its normal repulsion rate admits a point-wise non- degenerate maximum along M (0) among all locally C1-close material surfaces. We call M (t) an attracting LCS over [0,T] if it is a repelling LCS over [0,T] in backward time. We call M (t) a hyperbolic LCS over [0,T] if it is repelling or attracting LCS over [0,T]. With the aid of the following example we demonstrate the concept of the attracting and repelling surfaces. Example 15. We consider the system, − x˙ = x, y˙ = y. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: If

text: then

text: the

text: normal

text: growth

text: along

text: M

text: (t)

text: dominates

text: the

text: largest

text: tangen-

text: tial

text: growth

text: rate

text: along

text: M

text: (t).

text: From

text: the

text: point

text: of

text: view

text: of

text: computation,

text: both

text: the

text: normal

text: rate

text: and

text: ratio

text: can

text: be

text: computed

text: using

text: the

text: following

text: formulas

text: where,

text: With

text: the

text: aid

text: of

text: normal

text: repulsion

text: rate

text: and

text: ratio,

text: we

text: deﬁne

text: ﬁnite-time

text: repelling

text: (at-

text: tracting)

text: material

text: surfaces.

text: Deﬁnition

text: (Normally

text: repelling

text: material

text: A

text: material

text: surface

text: M

text: (t)

text: is

text: normally

text: repelling

text: over

text: time

text: interval

text: if

text: there

text: exist

text: numbers

text: α,β

text: such

text: that,

text: for

text: all

text: points

text: M

text: and

text: unit

text: normal

text: vector

text: we

text: have

text: eαt,

text: eβt.

text: A

text: material

text: surface

text: M

text: (t)

text: is

text: normally

text: attracting

text: over

text: time

text: interval

text: if

text: it

text: is

text: normally

text: repelling

text: over

text: in

text: backward

text: time.

text: We

text: call

text: M

text: (t)

text: hyperbolic

text: over

text: if

text: it

text: is

text: normally

text: attracting

text: or

text: normally

text: repelling

text: over

text: All

text: repelling

text: or

text: attracting

text: material

text: surfaces

text: are

text: not

text: equally

text: repelling

text: or

text: attracting.

text: Some

text: material

text: surfaces

text: are

text: more

text: repelling

text: or

text: attracting

text: than

text: others.

text: Material

text: surfaces

text: that

text: are

text: maximally

text: attracting

text: or

text: repelling

text: occupy

text: special

text: place

text: and

text: name

text: in

text: the

text: theory

text: of

text: normally

text: hyperbolic

text: invariant

text: manifolds.

text: They

text: are

text: called

text: as

text: Lagrangian

text: coherent

text: structures.

text: We

text: have

text: following

text: deﬁnitions

text: in

text: this

text: direction.

text: Deﬁnition

text: (Lagrangian

text: Coherent

text: Surface

text: Assume

text: that

text: M

text: (t)

text: is

text: a

text: normally

text: repelling

text: material

text: surface

text: over

text: t

text: We

text: call

text: M

text: (t)

text: a

text: repelling

text: Lagrangian

text: Co-

text: herent

text: Surface

text: (LCS)

text: over

text: if

text: its

text: normal

text: repulsion

text: rate

text: admits

text: a

text: point-wise

text: non-

text: degenerate

text: maximum

text: along

text: M

text: among

text: all

text: locally

text: material

text: surfaces.

text: We

text: call

text: M

text: (t)

text: an

text: attracting

text: LCS

text: over

text: if

text: it

text: is

text: a

text: repelling

text: LCS

text: over

text: in

text: backward

text: time.

text: We

text: call

text: M

text: (t)

text: a

text: hyperbolic

text: LCS

text: over

text: if

text: it

text: is

text: repelling

text: or

text: attracting

text: LCS

text: over

text: With

text: the

text: aid

text: of

text: the

text: following

text: example

text: we

text: demonstrate

text: the

text: concept

text: of

text: the

text: attracting

text: and

text: repelling

text: surfaces.

text: Example

text: We

text: consider

text: the

text: system,

text: x˙

text: x,

text: y˙

text: y.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##