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kastan & alon  @xcite identify and describe the configurations of relatively simple and small subnetworks that occur more frequently in biological networks than would be the case if the network were random . they refer to these subnetworks as _ network motifs_. later , in his 2007 book on systems biology  @xcite , alon makes the following comment    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` ideally , we would like to understand the dynamics of the entire network based on the dynamics of the individual building blocks . '' _ ( alon  @xcite . ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    the underlying premise behind this comment is that a modular , or engineering , approach to network dynamics is feasible . identify building blocks , connect together to form networks and then describe dynamical properties of the resulting network in terms of the dynamics of its components . this approach works well in linear systems theory , where a superposition principle holds , or in , for example , the study of synchronization in weakly coupled systems of nonlinear approximately identical oscillators where network dynamics can be closely related to the dynamics of individual nodes ( oscillators ) . however , from the perspective of a mathematician or physicist , alon s comment seems unhelpful for the study of dynamics of heterogenous networks modelled by systems of nonlinear odes . this is a well - known issue in complex systems  @xcite . yet engineers do couple gadgets together to make more complex systems and so it is natural to ask if it is possible to reconcile these viewpoints .    in this note , we outline some of the basic ideas involved in an approach to network dynamics based on what we call _ asynchronous networks_. we allow for features seen in networks from technology , engineering , and biology ( for example , neuroscience or gene transcription networks ) . network dynamics can involve a mix of distributed and decentralized control , adaptivity , event driven dynamics , switching , varying network topology and hybrid dynamics . typically network dynamics will be piecewise smooth , time - irreversible , nodes may stop and later restart , and there may be no intrinsic global time . significantly , many of these networks have a function : transportation networks bring people and goods from one point to another , neural networks may perform pattern recognition or computation , etc . our goal is to address alon s comment in the context of functional asynchronous networks . specifically , we describe a factorization of dynamics theorem where it is possible to describe the function of a large class of functional asynchronous networks in terms of the function of constituent subnetworks . as this article is only intended to be an introduction , we work always with the simplest examples and models and omit technical details . we refer the reader to  @xcite for greater detail , generality and proofs . we briefly describe some characteristic examples of asynchronous networks ( we refer to @xcite for more detail ) .      in threaded or parallel computation , computation is broken into blocks or ` threads ' which are then computed _ independently _ of each other at a speed that depends on the clock rates of the individual processors . as the computation proceeds , threads may need to exchange information with other threads . this process involves stopping and synchronizing ( updating ) the thread states . each thread may have to stop and wait for other threads to complete their computations before it can continue with its own computation . each thread has its own clock ( determined by its associated processor ) . if threads run on separate processors , threads will be unaware of the clock times of other threads except during the stopping and synchronization events . we give a simple detailed example of a transport network in section  [ sec : trans ] . in production networks , parts , compounds , etc . are repeatedly built and combined as part of a production process leading to the desired item ( for example , a car or protein ) . in particular , there is variation in both connection structure , typically intermittent , and in the set of nodes ( nodes may be combined or decomposed ) . a power grid consists of a network of various types of generators and loads connected by transmission lines . a microgrid is a local network , capable of existing independently of the main power grid , and typically powered by renewable energy sources ( for example , solar or wind power ) . critical questions involve the stability of the power grid in case of loss of a transmission line or generator ( variation in network structure ) , and control and stability issues related to combining and separation ( islanding ) of a large set of microgrids from the main power grid . many mathematical models from engineering and biology incorporate thresholds . for networks , when a node attains a threshold , there are often changes ( addition , deletion , weights ) in connections to another nodes . for networks of neurons , reaching a threshold can result in a neuron firing ( spiking ) and short term connections to other neurons ( for transmission of the spike ) . for learning mechanisms , such as spike - timing dependent plasticity ( stdp )  @xcite relative timings ( the order of firing ) are crucial  @xcite and so each neuron , or connection between a pair of neurons , comes with a ` local clock ' that governs the adaptation in stdp . in general , networks with thresholds , spiking and adaptation provide characteristic examples of asynchronous networks where dynamics is piecewise smooth and hybrid  a mix of continuous and discrete dynamics . we summarize some of the characteristic features of asynchronous networks as revealed in the examples above .    1 .   variable network structure and dependencies between nodes . changes depend on the state of the system or are given by a stochastic process . synchronization events associated with stopping or waiting states of nodes . 3 .   order of events may depend on the initialization of the system or stochastic effects . dynamics is only piecewise smooth and there may be a mix of continuous and discrete dynamics . aspects involving function , adaptation and control . 6 . evolution only defined for forward time systems are not time reversible . we use the notational conventions that @xmath0 and @xmath1 , @xmath2 . let @xmath3 . assume given a network with @xmath4 nodes , @xmath5 . suppose that @xmath6 has associated phase space @xmath7 , @xmath8 . set @xmath9  the network phase space . a vector field @xmath10 on @xmath11 is a _ network vector field_.    stopping , waiting , and synchronization are a feature of asynchronous networks . if nodes are stopped or partially stopped , node dynamics will be constrained to subsets of node phase space . we codify this situation by introducing a _ constraining node _ @xmath12 that , when connected to @xmath6 , implies that dynamics on @xmath6 is constrained . set @xmath13 . interactions between distinct nodes in the network are given by the network graph . connections @xmath14 encode _ dependencies _ if @xmath15 , and _ constraints _ if @xmath16 , @xmath8 . a _ connection structure _ @xmath17 is a directed network graph on the nodes @xmath18 such that for all @xmath8 , @xmath19 , @xmath20 , there is at most one directed connection @xmath14 . we do not allow self - loops or connections to the constraining node @xmath12 . @xmath17-admissible _ vector field @xmath21 is a network vector field with dependencies given by @xmath17 . if @xmath22 , @xmath23 , then @xmath24 does not depend on @xmath25 ( and conversely , see @xcite ) .    referring to figure  [ fig:1 ] , suppose @xmath26 is @xmath17-admissible . for @xmath27 , we have @xmath28 @xmath29 where here we have assumed that the constraint @xmath30 results in @xmath31 being stopped . a _ generalized connection structure _ @xmath32 is a ( nonempty ) set of connection structures on @xmath18 . an @xmath32-structure @xmath33 is a set @xmath34 of network vector fields such that each @xmath35 is @xmath17-admissible . suppose that @xmath32 is a generalized connection structure and @xmath33 is an @xmath32-structure . interactions between nodes in asynchronous networks may be state dependent or change over time ( stochastically ) . here we only consider state dependence . we handle interactions and constraints using an _ event map _ @xmath36 . the quadruple @xmath37 defines an _ asynchronous network_. dynamics on @xmath38 is given by the state dependent network vector field @xmath33 defined by @xmath39    we refer to  @xcite for the definition of an integral curve for ( [ eq:1 ] ) and note that the definition we use is different from that used in filippov systems  @xcite . under simple conditions on the event map  @xcite , it can be shown that if @xmath40 is compact then for each @xmath41 there is a unique piecewise smooth integral curve @xmath42 with initial condition @xmath43 and corresponding semiflow @xmath44 . in general , @xmath45 will not be continuous as a function of @xmath46 . we consider a single track railway line joining stations marked @xmath47 in figure  [ fig:2 ] . suppose there is a passing loop at @xmath48 . trains , marked @xmath49 and @xmath50 in figure  [ fig:2 ] , start from stations @xmath51 and @xmath52 respectively and proceed towards the opposite station . there is no central control or communication between the train drivers except when both trains are in the passing loop . we further assume that both drivers are running a nonlinear oscillator . when a train enters the passing loop it stops . when both trains are in the passing loop , the drivers cross couple their nonlinear oscillators . in order for a train to leave the passing loop , two conditions must be satisfied .    1 .   both trains must be in the passing loop . 2 .   the two nonlinear oscillators must be phase synchronized to within @xmath53 where @xmath54 . we model this setup using an asynchronous network with two nodes  corresponding to the two trains . the phase space for train @xmath55 will be @xmath56\times \mathbb{t}$ ] , @xmath57 ( @xmath58 ) , where the interval @xmath59 $ ] models the line joining @xmath51 to @xmath52 , @xmath51 has coordinate @xmath60 , @xmath52 has coordinate @xmath61 , the passing loop @xmath48 will be at @xmath62 , and @xmath63 will be the phase space for the nonlinear oscillator . assume train motion given by @xmath64 where @xmath65 @xmath66 $ ] . note that neither @xmath67 or @xmath68 can be zero anywhere on @xmath59 $ ] otherwise the trains will never both reach their destination stations in finite time . we define four connection structures . @xmath69    let @xmath70 be the associated generalized connection structure . model the uncoupled oscillator dynamics for train @xmath55 by @xmath71 , where @xmath72 , and the coupled dynamics by the kuramoto phase oscillator system @xmath73    it remains to define the admissible vector fields and event map that give the required dynamics for this example . as admissible vector fields ( on @xmath74 \times \mathbb{t})^2 $ ] ) we take @xmath75 we define the event map by @xmath76 finally , dynamics are given by the network vector field    @xmath77    we leave it as an easy exercise for the reader to check that if @xmath55 is initialized at @xmath78\times \mathbb{t}$ ] , @xmath79 , where @xmath80 , then ( [ eq:2 ] ) has a well defined integral curve @xmath81\times \mathbb{t})^2 $ ] , with specified initial condition , that gives the correct dynamics for the passing loop problem . the passing loop gives an example of a simple _ functional _ asynchronous network . the function is for the trains to go from one station to the opposite station in finite time . observe that for this example there is the possibility of a _ dynamical deadlock _ : if the trains start at the same time and if @xmath82 , then the coupled phase oscillators will never phase synchronize @xmath83 for all @xmath84  and so the trains will never exit the passing loop . we refer to  @xcite for more details on deadlocks in functional asynchronous networks . we follow the notational conventions of section  [ sec : async ] and let @xmath85 denote an asynchronous network . we assume that @xmath38 has associated semiflow @xmath86 suppose that we are given _ initialization _ and _ termination _ sets @xmath87 where @xmath88 typically , @xmath89 will be closed disjoint hypersurfaces that separate @xmath7 into three connected components , @xmath8 . that is , @xmath90 where @xmath91 and @xmath92 we call @xmath93 a _ functional asynchronous network_. the network function is getting from @xmath94 to @xmath95 and is expressed by the transition and timing functions @xmath96 that is , if @xmath97 , then for all @xmath8 there exists @xmath98 such that @xmath99    for the passing loop example discussed in the previous section , we take @xmath100 , @xmath101 . in this case , there is the implicit assumption that trains stop when they reach their termination set . equally well , we could take @xmath102 so that @xmath103\times\mathbb{t}$ ] etc . ( see also  @xcite ) . finally , observe that @xmath104 . more generally , we allow for general initialization times and define generalized transition and timing functions @xmath105 we refer to @xcite for details . for our main result , it is required that the network has a generalized transition and timing functions with @xmath106 .      in figure  [ fig:3 ] we show a nine node functional asynchronous network that is built from the eight `` events '' @xmath107 . the initialization and termination sets are indicated on the left and right sides of the figure respectively . the events signify regions of phase space where there can be ( state dependent ) interaction between nodes . for example , the event labelled @xmath108 involves interaction between nodes @xmath109 , @xmath110 , @xmath111 , and @xmath112 . observe that there is only a partially ordered temporal structure on the events . thus , the event @xmath108 must occur after @xmath113 but can occur before or after event @xmath114 .      in figure  [ fig:4 ] we represent a basic building block with the same number of inputs and outputs . the initialization sets are represented by the symbols @xmath115 , termination sets by @xmath116 . interaction between nodes occurs only in the event region denoted by the rectangle . outside of the event region , nodes evolve independently . more generally , we can allow for different number of inputs and outputs : nodes may merge or split . our immediate aim is describe some basic operations that we can define on functional asynchronous networks that allow us enable us to find a ( maximal ) decomposition of a functional asynchronous network into the form shown in figure  [ fig:3 ] . if @xmath117 , @xmath118 , are functional asynchronous networks with distinct node sets ( @xmath119 , @xmath120 ) , define the _ product _ @xmath121 to be the functional asynchronous network @xmath93 , where @xmath122 and @xmath123 is defined in the obvious way to be the asynchronous network with node set @xmath124 ( we refer to @xcite for details ) . we say the @xmath4-node functional asynchronous network @xmath93 is _ trivial _ if @xmath125 where each @xmath126 has exactly one node @xmath127 . in particular , if @xmath128 is trivial there are no interactions between nodes and no constraints . next let @xmath129 , @xmath118 , be a family of functional asynchronous networks with common initialization set , termination set and node set @xmath130 . suppose that for each @xmath131 , there exists @xmath132 such that    1 . @xmath133 where @xmath134 has node set @xmath135 and @xmath136 is trivial . 2 . if @xmath137 , @xmath138 . we define the _ amalgamation _ @xmath139 to be the functional asynchronous network @xmath140 , where @xmath141 is the trivial network defined as the product of the common trivial factors in @xmath136 , @xmath131 . thus the node set of @xmath141 will be @xmath142 .    referring to figure  [ fig:5 ] , we have @xmath143 and @xmath144 . the amalgamation @xmath145 is trivial when restricted to nodes @xmath146 . finally , we outline the operation of concatenation , referring the reader to  @xcite for the details ( most ) we omit . suppose that @xmath147 , @xmath148 , are functional asynchronous networks with common node set . assume that @xmath149 . the _ concatenation _ @xmath150 will be a temporal merging @xmath151 . we define    1 . @xmath152 , @xmath153 . 2 .   @xmath154 ,    where @xmath155 denote the join of the graphs . the definition of the set of admissible vector fields @xmath33 for @xmath38 is trickier and requires additional conditions on @xmath151  we refer to  @xcite for details . we define the event map by @xmath156 . we refer to figure  [ fig:6 ] for the operation of concatenation . the concatenation @xmath157 has the important property that if @xmath126 has generalized transition and timing functions @xmath158 , @xmath159 , @xmath160 , then @xmath157 has generalized transition function @xmath161 given by @xmath162  ( * ? ? ? * corollary 4.15 ) . we have deliberately avoided listing the detailed properties required of functional asynchronous networks in order to define amalgamations and concatenations . briefly , apart from requiring the existence of generalized transition and timing functions , we require ( 1 ) the uncoupled vector vectors defining intrinsic dynamics of a node @xmath6 to be transverse to @xmath163 , @xmath164 and ( 2 ) a local product structure on the network . we refer to  @xcite for the details . a functional asynchronous network is _ primitive _ if it can not be written as a nontrivial amalgamation or concatenation . [ fan ] under general conditions , a functional asynchronous network @xmath128 has a unique ( up to rearrangements ) decomposition @xmath165 where @xmath166 , @xmath167 , and and each @xmath168 is primitive . the generalized transition function @xmath161 for @xmath128 can be expressed in terms of the generalized transition and timing functions @xmath169 of @xmath170 ( or @xmath171 for @xmath168 ) by : @xmath172    consider the network shown in figure  [ fig:3 ] and assume that each event @xmath173 . @xmath174 is primitive . a decomposition satisfying the requirements of theorem  [ fan ] is indicated in figure  [ fig:7 ]  the dashed lines indicate the initialization and termination sets for the subnetworks .    the factorization for the network is @xmath175 this factorization corresponds to maximizing from the left hand side . however , if we maximize from the right we obtain the factorization @xmath176 in either case there is a concatenation of five networks  that is the minimum number possible . theorem  [ fan ] allows us to write the function of a network explicitly in terms of the transition functions of the constituent subnetworks . results of this type depend crucially on intermittent connection structure and nonsmooth dynamics . for example , no such result is possible for a classical coupled network of phase oscillators . the approach works because we have adopted an engineer s viewpoint : we emphasise function rather than dynamics . indeed , we are indifferent to the specific dynamics occurring _ between _ the initialization and termination sets . of course , both the timing and transition functions provide the key information about network function . theorem  [ fan ] is a prototypical theorem providing proof of concept . the conditions for the theorem can be significantly weakened from those required in  @xcite . the theorem yields maximal feedforward structure on a functional asynchronous network ( note that individual events may have feedback loops ) . 3 .   the result suggests the utility of starting with a small functional asynchronous network ; understanding the structure in depth and then then evolving to optimize function ( for example by adding feedback ) . there are many as yet unexplored issues such as bifurcation , hidden deadlocks , and the effects of noise . there is the problem of how far one can determine internal structure on the basis of input / output time series data .    99 u alon . _ an introduction to systems biology . design principles of biological circuits _ ( chapman & hall / crc , boca raton , 2007 ) . m di bernardo , c j budd , a r champneys , & p kowalczyk . _ piecewise - smooth dynamical systems _ ( springer , applied mathematical science 163 , london , 2008 ) . c bick & m j field . ` asynchronous networks and event driven dynamics ' , ( 2016 ) , preprint . c bick & m j field . ` asynchronous networks : modularization of dynamics theorem ' , ( 2016 ) , preprint . n caporale & y dan . ` spike timing dependent plasticity : a hebbian learning rule ' _ annu . neurosci . _ * 31 * ( 2008 ) , 2536 . a f filippov . _ differential equations with discontinuous righthand sides _ ( kluwer academic publishers , 1988 ) . w gerstner , r kempter , l j van hemmen & h wagner . ` a neuronal learning rule for sub - millisecond temporal coding ' , _ nature _ * 383 * ( 1996 ) , 7678 . w gerstner & w kistler . _ spiking neuron models _ ( cambridge university press , 2002 ) . n kashtan & u alon . ` spontaneous evolution of modularity and network motifs ' , _ pnas _ * 102 * ( 39 ) ( 2005 ) , 1377313778 . j ladyman , j lambert , and k wiesner . ` what is a complex system ? 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in this note we describe the theory of functional asynchronous networks and one of the main results , the modularization of dynamics theorem , which for a large class of functional asynchronous networks gives a factorization of dynamics in terms of constituent subnetworks . for these networks 
 we can give a complete description of the network function in terms of the function of the events comprising the network and thereby answer a question originally raised by alon in the context of biological networks .