Controlled Perturbation-Induced Switching in Pulse-Coupled Oscillator Networks
CControlled Perturbation-Induced Switchingin Pulse-Coupled Oscillator Networks
Fabio Schittler Neves and Marc Timme
Network Dynamics Group, Max Planck Institute for Dynamics and Self-Organization,Göttingen, D-37073, Germany. andBernstein Center for Computational Neuroscience (BCCN), Göttingen, Germany. ∗ Abstract
Pulse-coupled systems such as spiking neural networks exhibit nontrivial invariant sets in theform of attracting yet unstable saddle periodic orbits where units are synchronized into groups.Heteroclinic connections between such orbits may in principle support switching processes in thosenetworks and enable novel kinds of neural computations. For small networks of coupled oscillatorswe here investigate under which conditions and how system symmetry enforces or forbids certainswitching transitions that may be induced by perturbations. For networks of five oscillators wederive explicit transition rules that for two cluster symmetries deviate from those known fromoscillators coupled continuously in time. A third symmetry yields heteroclinic networks thatconsist of sets of all unstable attractors with that symmetry and the connections between them.Our results indicate that pulse-coupled systems can reliably generate well-defined sets of complexspatiotemporal patterns that conform to specific transition rules. We briefly discuss possibleimplications for computation with spiking neural systems.
PACS numbers: 05.45.Xt, 89.75.-k, 87.18.Sn ∗ [email protected] a r X i v : . [ n li n . AO ] N ov ntroduction Heteroclinic connections among saddle states are known to support non-trivial switchingdynamics in networks of units coupled continuously in time [1–5]. Interesting recent workfurthermore suggests that heteroclinic networks in state space may be used to encode a largenumber of spatiotemporal patterns if the transition between different states is controllable[6]. Supplementing such systems with certain additional features may thus enable a newkind of computation [7].Networks of pulse-coupled oscillators, that model, e.g., the dynamics of spiking neuralnetworks, constitute hybrid systems that are very distinct from systems coupled continuouslyin time. In pulse-coupled hybrid systems pulses interrupt the otherwise smooth timeevolution at discrete event times when pulses are sent or received. Such networks mayexhibit unstable attractors [8], unstable saddle periodic orbits that are attractors in thesense of Milnor [9]. Recent works indicate that unstable attractors may generically occurin systems with symmetry [10, 11] and that such saddle periodic orbits may be connectedto heteroclinic networks in a standard way, but with some anomalous features [12, 13].In particular, due to the attractor nature of the periodic orbits, switching among saddlesrequires external perturbations. It was known before that in non-hybrid systems with non-attracting saddles, such perturbations may in principle direct the switching path. In thiswork we study under which conditions and precisely how small controlled perturbationscan exploit heteroclinic connections in pulse-coupled systems to support switching processesamong saddle states, a key prerequisite for computation by heteroclinic switching.The results may be of particular relevance for neural systems where pulses are electricaction potentials (spikes) generated by neurons because spatiotemporal switching patternsof spikes have been suggested to underly information processing [14, 15].This article is divided into four main sections. After introducing the model and explainingour analytical approach in the first section, in the second we present the most persistentattractors and their symmetries. In the third section, we derive the dynamic response of thesystem to single oscillator perturbations and provide a local stability analysis. Finally, weconclude discussing the relation between switching processes in the pulse-coupled systemsconsidered to those in systems coupled continuously in time. We also briefly discuss potentialimplications for neural coding and paths to future investigations.
I. Pulse-coupled network
Consider a network of N oscillators that are connected homogeneously all-to-all withoutself-connections through delayed pulse-couplings. The state of each oscillator i ∈ { , . . . , N } at time t is specified by a single phase-like variable φ i ( t ) [16]. In the absence of interactions,its dynamics is given by dφ i dt = 1 , ≤ φ i ≤ . (1)When oscillator i reaches a threshold, φ i ( t − ) = 1 , its phase is reset to zero, φ i ( t ) = 0 , andthe oscillator is said to send a pulse. Such pulse is sent to all other oscillators which receivethis signal after a delay time τ . The incoming signal induces a phase jump φ i ( t ) = H (cid:15) ( φ i ( t − )) = U − (cid:2) U (cid:0) φ i ( t − ) (cid:1) + (cid:15) (cid:3) , (2)2hich depends on the instantaneous phase φ i ( t − ) of the post-synaptic oscillator and theexcitatory coupling strength (cid:15) > . The phase dependence is determined by a twicecontinuously differentiable potential function U ( φ ) that is assumed to be strictly increasing( U (cid:48) ( φ ) > ), concave down ( U (cid:48)(cid:48) ( φ ) < ), and normalized such that U (0) = 0 , U (1) = 1 . Asshown in [8, 17], this phase dynamics is equivalent to the ordinary differential equations dV i dt (cid:48) = f ( V i ) + S i ( t (cid:48) ) , (3)where S i ( t (cid:48) ) = N (cid:88) j =1 j (cid:54) = i (cid:88) k ∈ Z (cid:15)δ (cid:16) t − τ (cid:48) − t (cid:48) jk (cid:17) , (4)is a sum of delayed δ -currents induced by presynaptic oscillators. Oscillator j sends its k thpulse at time t (cid:48) jk whenever its phase variable crosses threshold, V j ( t (cid:48) − jk ) ≥ ; thereafter, itis instantaneously reset, V j ( t (cid:48) jk ) → . The k th pulse of oscillator j is received by i after adelay τ (cid:48) . The positive function f ( V ) > yields a free ( S i ( t (cid:48) ) ≡ ) solution V i ( t (cid:48) ) := V ( t (cid:48) ) = V ( t (cid:48) + T ) of intrinsic period T . The above function U ( φ ) is related to this solution via U ( φ ) = V ( φT ) , (5)defining a natural phase φ by rescaling the time axis, t = t (cid:48) /T and τ = τ (cid:48) /T . We focus on the specific form U ( φ i ) = U IF ( φ i ) = I i γ (cid:0) − e − φ i T IF (cid:1) that represents theintegrate-and-fire oscillator defined by f ( V ) = I − γV . Here I > is a constant externalinput and T IF = γ log (cid:0) − γI (cid:1) − the intrinsic period of an oscillator. Any U ( φ ) sufficientlyclose to U IF ( φ ) give qualitatively similar results.After defining the dynamics of the network elements we can define its collective state asa phase vector, φ = ( φ , φ , . . . , φ N ) , (6)where each φ i describes the phase of oscillator i . Their dynamics is governed by (1) and (2).The difference in phase among them will define the macroscopic states of the network, asexplained in the next section.The event-based updating presented above brings two main advantages: It yields exactanalytical solutions of state space trajectories and substantially reduces the simulation timecompared to numerical integration with fixed small time steps. II. Periodic orbit dynamic and symmetries
Here we define and explicitly study the dynamics of partially synchronized states, periodicorbits where groups of oscillators are identically synchronized into clusters, for three mainsymmetries of N = 5 oscillators. The analysis reveals mechanisms of perturbation-inducedswitching transitions that critically depend on the local stability properties of cluster periodicorbits. As we show below, stability in turn is determined by whether a cluster receives onlysub-threshold input during one period (“unstable” cluster) or it also receives supra-thresholdinput (“stable” cluster), the only two options available. Thus, similar switching mechanismsfor a given symmetry will prevail also for larger N , cf [17], and contribute to much more3omplex saddle state transitions, cf figure 6. As shown in the last section, when a constantexternal input I to a single oscillator i is sufficiently strong to drive the membrane potentialto cross its threshold ( U (cid:48) > ), the potential dynamics becomes periodic with period T .It was known before that networks of such pulse-coupled oscillators may exhibit differentinvariant states including partially synchronized states [5, 8, 9, 18, 19].To explore the possible unstable attractors we systematically varied the parameters andthe initial conditions for our system and found numerically that three clustered states presentthose state symmetries most persistent to perturbations. Two of these states are composedof two clusters, with permutation symmetries S × S and S × S , respectively; another one iscomposed of two clusters and one single element, with permutation symmetry S × S × S .The event-based analyses of these states is based in return maps that are presented indetail in tables I, III, IV and VI. For each of these three cluster periodic orbits, the eventsequence of sending and reception of pulses fully defines the type of periodic orbit such thatthe analytical conditions for existence of a family of such orbits can be directly read fromthese tables. In particular, these three families of periodic orbits exist for an open set ofparameters close to the three examples numerically specified in tables II, V and VII. Theexistence conditions for each periodic orbit naturally imply that the phases of all oscillatorsexactly return to the same value after a fixed period; at the same time, the predefined eventsequence must be kept.Throughout this work, we represent the dynamical states relative to the symmetries S × S , S × S and S × S × S , respectively, by the phase vectors φ = ( a, a, a, b, b ) , (7a) φ = ( a, a, a, a, b ) , (7b) φ = ( a, a, b, b, c ) , (7c)where each element represents one oscillator and the letters indicate to which cluster itbelongs. In these periodic orbits the differences in phase ( a − b ) , ( a − c ) and ( b − c ) willchange in time in a periodic manner, while the cluster configuration remains the same. Inthis notation the elements labeled as ’a’ belong to an unstable cluster, as ’b’ to a stablecluster, while ’c’ represents a single element that reacts stably to small perturbations (seebelow).It is important to emphasize that this system exhibits symmetric connections and thatthe parameters, I , γ , (cid:15) and τ , are global (see section I). As a consequence, the initialcondition controls the final attractor and determines which of the permutation-equivalentstates is obtained. By the same symmetry argument, the number of permutation-equivalentconfigurations for each state symmetry is given by the number of ways we can form thevectors presented above, which results in 10, 5, and 30 states, respectively. In the nextsection we study the stability of these cluster states and the possible state transitions amongthem in the presence of small perturbations. III. Stability and switching properties
In this section we will study, case by case, the dynamics and stability of cluster periodicorbits presented in the last section. First we show that these periodic orbits actually areunstable attractors, and later we study the possible transitions between different states in4esponse to small perturbations.To study the local stability of these attractors, we introduce a perturbation vector, δ ( n ) = ( δ ( n ) , δ ( n ) , δ ( n ) , δ ( n )) , (8)that has four components since only the relative phases among the oscillators are relevant( δ ( n ) ≡ ). The analysis presented here consists of a study of the temporal evolution ofthis perturbation vector at each cycle. Thus, δ i ( n ) := φ i ( t ,N ) − φ ∗ i ( t ) (9)are the perturbations to phases on the periodic orbit just after oscillator one has sent its n th pulse and been reset, i.e. δ i ≡ .After a small enough initial perturbation that is added to the phase vector at some pointof the unperturbed dynamic, the temporal evolution of the perturbation vector is definedas the difference between this perturbed vector after one cycle of the system dynamic andthe unperturbed phase vector at the same time. Analytically tracking the periodic orbitdynamics (cf tables I, IV, and VI) yields the perturbation perturbation vector after onecycle as a function of the perturbation in the previous cycle, δ ( n + 1) = F ( δ ( n )) , (10)which can be linearly approximated by δ ( n + 1) . = J δ ( n ) , (11)where J is the Jacobian matrix at δ ( n ) = 0 , describing the local dynamics.After analyzing the local stability properties, we study non-local effects in response tosingle oscillator perturbations. The procedure consists of perturbing only one oscillator ateach time. We consider negative perturbations, instantaneous decrements on the phase, andpositive ones, instantaneous increments on the phase. When possible, transition diagramsare included. A. Clustered state S × S For the clustered state S × S we have the temporal evolution for the vector δ in onecycle, (cf tables VIII and I) assuming δ < δ , and δ < δ , given by: δ ( n + 1) = φ − (0 , , A, A ) (12)where φ is the phase vector given by the last row of table VIII and the vector (0 , , A, A ) represents the unperturbed cycle (see table I). By (11) and (12), we obtain the followingJacobian matrix: ∂ δ ( n + 1) ∂ δ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) δ ( n )=0 = α j β γ j γ j , (13)where α , β , and γ are positive reals larger than one, j and j are positive, and j is muchsmaller than the other elements (for analytical expressions of the partial derivatives refer toA). This matrix has two zero eigenvalues with eigenvectors that correspond to the directions5 IG. 1.
Example of perturbation-induced switching in a S × S state set. The response of the system to asequence of five negative single oscillator perturbations preserving a S × S clustered symmetry ( τ = 0 . , (cid:15) = 0 . , I = 1 . , γ = 1 ). The phases of all oscillators are plotted at the moment when oscillator 1 isreset, each color representing the phase of one oscillator. There are transitions through two steps, wherein a first moment the cluster S is unstable, after one perturbation (as shown by the second and fourthperturbation) it reaches a new configuration with the cluster S in a unstable phase position, a secondperturbation (as shown in the first, third and fifth perturbations) is needed to put the system in the initialphase difference again, maintaining the cluster components but changing its stability. The symmetry ofthe unstable attractors are preserved. The sequence of states given by the plateaus are ( a, a, b, b, a ) ∗ → ( a, a, b, b, a ) → ( b, b, a, a, a ) ∗ → ( b, b, a, a, a ) → ( a, a, a, b, b ) ∗ → ( a, a, a, b, b ) , where the star indicate thestates where S is in a unstable phase. of δ and δ ; and two non-zero eigenvalues given by, λ = α = (cid:104) − (cid:104) H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( H (cid:15) ( τ )) (cid:105) H (cid:48) (cid:15) ( τ + H (cid:15) ( τ )) (14) λ = β = (cid:104) − H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( H (cid:15) ( τ )) H (cid:48) (cid:15) ( τ + H (cid:15) ( τ )) , (15)where H (cid:48) (cid:15) ( φ ) = ∂∂φ U − ( U ( φ ) + (cid:15) ) . (16)Here λ and λ are larger than one (see Lemma 1), noticing that all terms are due tosub-threshold events. Thus a perturbation can effectively disturb the system in two differentpossible directions, showing that the cluster S is stable and the S is unstable. Lemma 1. If H (cid:15) ( φ ) given by (2) mediates a sub-threshold reception event and (cid:15) > , U (cid:48) ( φ ) > , and U (cid:48)(cid:48) ( φ ) < , then H (cid:48) (cid:15) ( φ ) > . roof. Assume (cid:15) > . By definition H (cid:48) (cid:15) ( φ ) = ∂∂φ U − ( U ( φ ) + (cid:15) ) = U (cid:48) ( φ ) U (cid:48) ( U − ( U ( φ ) + (cid:15) )) = U (cid:48) ( φ ) U (cid:48) ( H (cid:15) ( φ )) , Since U (cid:48) ( φ ) is a monotonic decreasing function and H (cid:15) ( φ ) > φ we have U (cid:48) ( φ ) > U (cid:48) ( H (cid:15) ( φ )) for any H (cid:15) ( φ ) , and consequently H (cid:48) (cid:15) ( φ ) > .Now we describe the long-term effect of a single oscillator perturbation to the unstablecluster S . A negative perturbation to one of the elements on the unstable cluster ( φ + =( a, a, a − δ , b, b ) ) puts one of its elements phase slightly behind; then the initial stablecluster S begins to receive an additional pulse just after it is reset, increasing its relativephase in each cycle, and thus approaching the phase of the elements in the originally stablecluster. After some cycles it finally joins that cluster by a simultaneous reset, forming anew S × S clustered state. This switching process is illustrated just after the second andfourth perturbations in figure 1. The final state has the same symmetry as the initial state,but has different stability properties: Whereas the orbit is stable to splitting the S cluster,it is unstable to splitting the S cluster and upon perturbation resynchronizes and shifts inphase with respect to the cluster S (see table III). A further perturbation to the cluster S does not change the elements of each cluster but just returns the system to the initial phasedifference, as illustrated in the first, third and fifth perturbations in figure 1.Intriguingly, positive perturbations φ + = ( a, a, a + δ , b, b ) ) result in a completelydifferent dynamic, as can be seen in figure 2 which presents a sequence of three negativeand two positive perturbations. A positive perturbation puts just one oscillator from theunstable cluster ahead, that now increases its phase in relation to its original cluster ineach cycle till it begins to be reset by pulses coming from the originally stable cluster. The FIG. 2.
Example of symmetry change by perturbation-induced switching. A sequence of three negativeand two positive single oscillator perturbations (same parameters as in figure 1). Showing that the positiveperturbations splits the unstable cluster until the system reach some periodic orbit. The symmetry is notpreserved. S cluster changes its phase to conform with this new pulse configuration, but stillbeen reset by pulses coming from the two elements left on the unstable cluster. Thus the S cluster splits into two clusters, and the new configuration becomes S × S × S . A furtherperturbation puts the system in a stable cyclic state.Hence the symmetry S × S is not preserved upon a general perturbation. however,simulations suggest that if we only consider negative single oscillator perturbations, or onenegative perturbation with a larger magnitude than the others, the symmetry is preserved,and it is possible to write a transition rule, ( a − δ , a, a, b, b ) → ( a, b, b, a, a ) , (17)that permits us to know which will be the next state after one perturbation B. Clustered state S × S Considering now the S × S symmetry, assuming δ < δ < δ and δ > , in an procedureanalogous to that in the last section, we obtain the following Jacobian matrix (see last rowof tables IX and IV): ∂ δ ( n + 1) ∂ δ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) δ ( n )=0 = α j β j j γ j j θ , (18)here α , β , γ , and θ are larger than one, j , j are positive, and j , j , and j are muchsmaller than the other elements (see A). This matrix has one zero eigenvalue, correspondingto the single element represented by S and three non-zero eigenvalues given by λ = α = (cid:104) − (cid:104) H (cid:48) (cid:15) ( τ ) H (cid:48) (cid:15) ( H (cid:15) ( τ )) (cid:105) H (cid:48) (cid:15) ( H (cid:15) ( τ )) (cid:105) H (cid:48) (cid:15) ( τ + H (cid:15) ( τ )) , (19) λ = β = (cid:104) − (cid:104) H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( H (cid:15) ( τ )) (cid:105) H (cid:48) (cid:15) ( H (cid:15) ( τ )) H (cid:48) (cid:15) ( τ + H (cid:15) ( τ )) , (20) λ = γ = (cid:104) − H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( H (cid:15) ( τ )) H (cid:48) (cid:15) ( H (cid:15) ( τ )) H (cid:48) (cid:15) ( τ + H (cid:15) ( τ )) . (21)Using the same argument as in the last section (see lemma 1), these three eigenvalues arenecessarily larger than one ( λ , λ , λ > ), showing the instability of the cluster S .Again we study the effect of single oscillator perturbations. As can be seen in figure 3,positive perturbations to an element of the unstable cluster S put one element from theunstable group ahead. This difference in phase will increase each cycle, since pulses receivedat larger phases more strongly shift the oscillator’s phase; at the same time the S element isnot only reset by the other oscillators pulses, but also receives additional pulses that makesits phase approach the unstable cluster. After some cycles, the element S joins the originalunstable cluster, forming a new S cluster while the perturbed oscillator forms the new S , returning the system to its original phase difference and symmetry, but with differentelements composing the clusters.When a negative perturbation is applied (see figure 4) the element perturbed is putbackwards, and as before the elements ahead increase the difference in phase in relation tothe perturbed element; at each cycle, after being reset, the original S element receive anadditional pulse coming from the perturbed element, increasing its phase. After some cyclesthis increase makes the S element to join the perturbed element, forming a new cluster S .The new configuration becomes S × S , where S is unstable. A second perturbation to the8 IG. 3.
Example of perturbation-induced switching in a S × S state set. A sequence of four positivesingle oscillator perturbations preserving a S × S clustered state ( τ = 0 . , (cid:15) = 0 . , I = 1 . , γ = 1 ).The phase of all oscillators are plotted each time oscillator 1 is reset, each color represents the phase of oneoscillator. The perturbed oscillator leaves the cluster S and replaced the S oscillator, that join the cluster S , preserving the symmetry. The sequence of states corresponding to the plateaus are ( a, a, a, a, b ) → ( a, b, a, a, a ) → ( a, a, b, a, a ) → ( a, a, a, b, a ) . FIG. 4.
Example of symmetry change by perturbation-induced switching. A sequence of three negativesingle oscillator perturbations for the same parameter used on figure 3. The perturbed oscillator joins the S oscillator, forming a S cluster, new perturbations led first to an S × S configuration and later to splitthe clusters reaching a stable attractor. The symmetry is not preserved. cluster moves the perturbed element to the cluster S . A last perturbation can either putthe system in its initial configuration or split the cluster into two, depending on the positionon the periodic orbit it is applied. The symmetry of this state is obviously not preserved fornegative perturbations, and computer simulations indicate that more general perturbationsbring an even more complicated switching dynamic due the large number of elements in theunstable cluster.Considering only single positive perturbations, we can state a transition rule between twostates when subject to a single positive perturbation, ( a + δ , a, a, a, b ) → ( b, a, a, a, a ) . (22)Under these considerations, the resulting transition diagram is a fully connected one, and itis possible to jump from one equivalent permutation state to any other one applying onlyone perturbation. C. Clustered state S × S × S For the symmetry S × S × S , assuming δ < δ and δ , δ > , we have the followingJacobian matrix: ∂ δ ( n + 1) ∂ δ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) δ ( n )=0 = α β β γ , (23)where α > β > γ > (see A), cf tables X and IV. This matrix has three zero eigenvaluesand only one non-zero eigenvalue given by λ = α = H (cid:48) (cid:15) (cid:16) τ (cid:48) + H (cid:15) (cid:16) τ − τ (cid:48) + H (cid:15) ( τ (cid:48) ) (cid:17)(cid:17) (cid:104) − H (cid:48) (cid:15) (cid:16) τ − τ (cid:48) + H (cid:15) ( τ (cid:48) ) (cid:17) (cid:104) H (cid:48) (cid:15) ( τ (cid:48) ) (cid:105)(cid:105) , (24)which is larger than one accordingly to Lemma 1. The fact that there is only one eigenvalueand that it is larger than one not only shows that there is only one unstable cluster, butalso that perturbations change only the difference in phase between the two elements onthis cluster. As a result, any general perturbation can be mapped to a single oscillatorperturbation.Differently from the last two considered symmetries, we here have one single element,one stable S clusters, and one symmetric unstable S cluster. When perturbed, the initialunstable cluster S splits into two, the additional pulse received now by the initial single S element just after its reset makes it approaches the element that was put behind on theunstable S cluster, forming a new stable S cluster. This occurs because it is reset bysupra-threshold pulses. Moreover the element ahead begins to be reset by pulses and stopsincreasing its phase, becoming stable and the original stable S cluster after changing itsphase, is not reset by pulses anymore, becoming unstable. The final state has the samesymmetry and stability properties as the former state.The preservation of the symmetry implies a closed transition diagram among all thepossible S × S × S states (see figure 6). We state two simple equivalent switching rules.Considering first a positive representation we have ( a, a + δ , b, b, c ) → ( c, b, a, a, b ) (25)10 IG. 5.
Example of perturbation-induced switching in a S × S × S state set. A sequence of fiveperturbations driving the system through different states with symmetry S × S × S ( τ = 0 . , (cid:15) =0 . , I = 1 . , γ = 1 ). The phase of all oscillators are plotted at the moment when oscillator number oneis reset, each color representing the phase of one oscillator. The apparent change of the phase differencesamong the clusters just after the perturbations depends on the cluster to which the reference oscillatorbelongs. The symmetry of the unstable attractors is preserved. The sequence of states corresponding to theplateaus are ( a, a, b, b, c ) → ( b, c, a, a, b ) → ( a, b, b, c, a ) → ( b, a, a, b, c ) → ( a, c, b, a, b ) → ( b, b, a, c, a ) . that can be rewritten for negative perturbations simply as ( a − δ , a, b, b, c ) → ( c, b, a, a, b ) . (26)We conclude that for this symmetry the unstable attractors are linked to form aheteroclinic network (figure 6), characterized by (25) and (26), forming a closed set ofsaddle periodic orbits among which the systems switches in a controlled way upon smallexternal perturbations. We remark that in the absence of noise this dynamic does notexhibit spontaneous transitions between nearby saddle states [12, 13] but instead displaysconvergence to unstable attractors. The free dynamic of each element evolves continuouslyup to reset; still, the collective dynamic of the entire network (network state) evolvescontinuously almost always, but interrupted by discrete jumps due to the infinitely fastphase response of the interaction. As (i) the transitions are fully controlled by externalperturbations and thus predictable, and (ii) the symmetry is preserved when the networkis subject to sufficiently small, general perturbations, arbitrarily small external noise wouldtrigger a persistent switching dynamic in which the network states are constrained to theclosed set of periodic orbits with initial S × S × S symmetry. A numerical example of thisspontaneous switching phenomenon has been reported before [17] for a system of N = 100 oscillators exhibiting S × S × S × S × S symmetry.11 IG. 6. Five steps state transition diagram for the symmetry S × S × S . This diagram showsall the possible 5 steps paths, sequence of states, beginning on the state {a,b,c,b,a}. Each arrowcorrespond to one of the two possible perturbations, subject to (25) and (26). It is necessary atleast 5 perturbations to reach the initial state again. IV. Discussion
In the networks of pulse-coupled oscillators studied above, three sets of heteroclinicallyconnected unstable attractors appear to have a well-defined symmetry that depends onthe network parameters. Interestingly, for two state symmetries, the possible switchingtransitions markedly deviate from those in time-continuously coupled systems [20]. Moreover,all attractors with the third symmetry S × S × S form one closed heteroclinic network,where all possible transitions are predictable and depend on the precise direction of theperturbation. In fact, mapping an arbitrary small perturbation to a single oscillatorperturbation, we derived a general set of transition rules, (25) and (26). This last featureguarantees that there are no changes of symmetry during the switching and precisely definesa transition diagram (figure 6) that holds for all sufficiently small perturbations.Thus, this work explicitly shows how nontrivial switching dynamics is induced andprecisely controlled by perturbations in pulse-coupled systems. Our analysis shows thatand how event sequences, collectively generated by the network, fully determine switchingtransitions in pulse-coupled oscillators. As a consequence, these results are not restrictedto the IF model (used here for numerical simulations and illustrating purposes) but equallyhold for different oscillator models with sub-threshold potential dynamics that are sufficientlyclose to the one considered here. Moreover, the phenomenon should still hold qualitativelyfor temporally extended responses as long as the post-synaptic response times are shortcompared to the membrane time constant and inter-spike-interval times. Nevertheless,although we expect the same transition possibilities, the dynamics without noise will showheteroclinic switching sequences that depend on the initial condition and do not requireexternal perturbations, cf [13]. As stability and instability of clusters reflect synchronizingand desynchronizing mechanisms [19], here realized by supra- and sub-threshold inputs,respectively, similar switching features also occur in networks of N ≥ pulse-coupled12scillators [17].Since the systems studied here are pulse-coupled and of hybrid type, with smooth timeevolution interrupted at discrete times of interactions, it is interesting to compare ourresults to those on systems of oscillators coupled continuously in time [6, 7]. The lattersystems exhibit partially synchronized saddle states with the same symmetry S × S × S ,where a persistent switching dynamics appears as one feature of the model and, whensubject to asymmetric external currents, generates a wide variety of spatiotemporal patterns.Interestingly, the transitions rules given by (25) and (26) (and illustrated on figures 5and 6) not only guarantee an equivalent persistent switching dynamic when subject tonoise, but also imply as well the existence of the same set of spatiotemporal patternswhen subject to asymmetric external currents. Thus, our model characterizes exactlythis switching dynamic in a pulse-coupled neuronal framework, where the patterns canbe described as distributed pulse-sequences (spike patterns). The importance of such aspiking representation becomes evident in particular when considering potential applicationsto neural coding and information processing [7]. For instance, studies on the olfactory systemof insects [21, 22] have shown that biological systems could use spatiotemporal spike patternsas part of their information processing. In particular our results agree with the interestingpredictions of Hansel et al [2], Rabinovich et al [3], and Timme et al [8] regarding thegeneration of spatiotemporal spike patterns based on a switching dynamics. In addition ourwork presents a neural system where the entire (long-time) switching dynamics follows froma fixed set of transition rules, a promising feature that may prove not only advantageous forcomputation in biological but also in artificial systems.We remark that although the apparent equivalence between the dynamics of pulse-coupledoscillators and continuously coupled oscillators works for this specific symmetry, it does notholds as a general rule. The most pronounced counter-examples are systems with symmetry S × S , which when smoothly coupled exhibit persistent switching dynamics, but whenpulse-coupled, any continuous small noise required for the switching necessarily drives thesystem to a stable attractor, cf figure 2.To understand how these switching properties may actually perform computational tasks,a complete analysis of the effect of asymmetric currents, driving pulses and asymmetricconnections on the spike patterns is needed. The answer to these controlling factors couldbring us important information about alternative mechanisms of neural computation, bothbiological and artificial. Acknowledgments
We thank Frank van Bussel for critically checking the English in this manuscript. Wethank the Federal Ministry of Education and Research (BMBF) Germany for support undergrant number 01GQ0430 to the Bernstein Center for Computational Neuroscience (BCCN)Göttingen.
A. Partial derivatives
Here we present the analytical expressions for the Jacobian matrices presented on sectionII. Where we introduce a short notation H y ( x ) → x y , for x ∈ (cid:8) , τ, ( τ − τ (cid:48) ) , ( τ − τ (cid:48) + τ (cid:48) y ) (cid:9) and y ∈ { (cid:15), (cid:15), (cid:15) } . S × S : non-zero elements for the Jacobian matrix (13) α = (cid:104) − (cid:104) H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A1) β = (cid:104) − H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A2) γ = (cid:104) − H (cid:48) (cid:15) (0) (cid:105) H (cid:48) (cid:15) ( τ + 0 (cid:15) ) + H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A3) j = − (cid:104) (cid:104) − H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A4) j = j = (cid:104) − H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A5) S × S : non-zero elements for the Jacobian matrix (18) α = (cid:104) − (cid:104) H (cid:48) (cid:15) ( τ ) H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A6) β = (cid:104) − (cid:104) H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A7) γ = (cid:104) − H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A8) j = j = − (cid:104) (cid:104) − H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A9) j = − (cid:104) − H (cid:48) (cid:15) (0 (cid:15) ) (cid:105) H (cid:48) (cid:15) (0 (cid:15) ) + (cid:104) − H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A10) j = − (cid:104) − H (cid:48) (cid:15) (0) (cid:105) H (cid:48) (cid:15) (0 (cid:15) ) H (cid:48) (cid:15) (0 (cid:15) ) + (cid:104) − H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A11) θ = − H (cid:48) (cid:15) (0) H (cid:48) (cid:15) (0 (cid:15) ) H (cid:48) (cid:15) (0 (cid:15) ) + (cid:104) (cid:104) − H (cid:48) (cid:15) ( τ ) (cid:105) H (cid:48) (cid:15) ( τ (cid:15) ) H (cid:48) (cid:15) ( τ (cid:15) ) (cid:105) H (cid:48) (cid:15) ( τ + τ (cid:15) ) (A12) S × S × S : non-zero elements for the Jacobian matrix (23). λ = α = H (cid:48) (cid:15) (cid:16) τ (cid:48) + (cid:16) τ − τ (cid:48) + τ (cid:48) (cid:15) (cid:17) (cid:15) (cid:17) (cid:104) − H (cid:48) (cid:15) (cid:16) τ − τ (cid:48) + τ (cid:48) (cid:15) (cid:17) (cid:104) H (cid:48) (cid:15) ( τ (cid:48) ) (cid:105)(cid:105) , (A13) β = H (cid:48) (cid:15) (0) H (cid:48) (cid:15) ( τ (cid:48) + 0 (cid:15) ) + H (cid:48) (cid:15) (cid:16) τ (cid:48) + ( τ − τ (cid:48) + τ (cid:15) ) (cid:15) (cid:17) (cid:104) − H (cid:48) (cid:15) ( τ − τ (cid:48) + τ (cid:48) (cid:15) ) (cid:105) (A14) γ = − (cid:104) − H (cid:48) (cid:15) ( τ − τ (cid:48) ) (cid:105) H (cid:48) (cid:15) (cid:16) ( τ − τ (cid:48) ) (cid:15) (cid:17) + H (cid:48) (cid:15) (cid:16) τ (cid:48) + ( τ − τ (cid:48) + τ (cid:48) (cid:15) ) (cid:15) (cid:17) (cid:104) − H (cid:48) (cid:15) ( τ − τ (cid:48) + τ (cid:48) (cid:15) ) (cid:105) (A15) B. Return Maps
Here we explain step by step the periodic orbit dynamic described by the unperturbedreturn maps that define the three main families of attractors, given in tables I, III, IV andVI. The event notation is the following: s i indicates that oscillator i sent a pulse; r i indicatesthat pulses were received coming from the oscillators indicated by i . Capital letters indicate14onstants, H (cid:15) ( φ ) is the transfer function presented on section 1, and p i,j is a short notationfor the phase of oscillator i at event number j .A realization of the dynamics described by these tables are presented for specificparameter in tables II, V and VII. As there are no approximation to the correspondinganalytical condition tables, the specific values completely agree with the iterated simulationin figures 1, 3 and 5. Unperturbed S × S dynamic The initial condition is such that no pulse was sent before time zero. At time zero, thefirst event, oscillators 1, 2 and 3 fire ( s ); the second event is the reception of these signalsa time τ later ( r ), these is an supra-threshold event to oscillators 4 and 5, which thensend a signal ( s ) and are reset; the third event is the reception of pulses from 4 and 5( r ); and the last event is the reset of oscillators 1, 2 and 3 ( s ) by reaching the threshold.For any choice of the parameters resulting in A = p , + 1 − p , (while preserving the eventsequence), we have a period-one attractor, since the initial state is obtained after one pulseof each oscillator. A numerical example of such a structure is presented in table II. Fromthis map, we can conclude that the cluster S is stable, since any small variation will berestored when its elements are reset together by the incoming pulse. TABLE I. Analytic table of condition for an unperturbed S × S dynamic, S unstable. event time φ , φ , φ φ , φ event num. s , , r , , ; s , τ H (cid:15) ( τ ) = p , H (cid:15) ( A + τ ) > → r , τ H (cid:15) ( p , + τ ) = p , H (cid:15) ( τ ) = p , s , , τ + 1 − p , → p , + 1 − p , TABLE II. Analytic prediction of phase dynamic for parameters τ = 0 . , (cid:15) = 0 . , I = 1 . and γ = 1 , realizing a S × S cycle. event time φ , φ , φ φ , φ event num. s , , r , , ; s , r , s , , Unperturbed S × S dynamic, S unstable This map describe the partner orbit of table I, once they appear for the same range of theparameter, but for different initial conditions. The initial condition here is that pulses fromoscillators 4 and 5 are received ( r (4 , ) exactly at time 0, forcing oscillators 1, 2 and 3 to fire( s , , ), what define the first event; the second event is the reception of these pulses ( r , , ) τ time later; the third event is the reset of oscillators 4 and 5 upon reaching the threshold( s , ), and consequently the generation of two new pulses; the last event is the reception ofthese pulses ( r , ), which causes oscillators 1, 2 and 3 to generate one pulse ( s , , ). In this15ase the S cluster is stable, since any small variation on the phase of its components willdisappear in the next cycle when all are reset together by incoming pulses [10]. TABLE III. Analytic table of condition for an unperturbed S × S dynamic, S unstable. event time φ , φ , φ φ , φ event num. r (4 , ; s , , r , , τ H (cid:15) ( τ ) = p , H (cid:15) ( C + τ ) = p , s , − p , p , − p , = p , → r , ; s , , τ + 1 − p , H (cid:15) ( p , + τ ) > → H (cid:15) ( τ ) Unperturbed S × S dynamic This map describes another period-one attractor, where no pulse was sent before time 0.The first event is the signal sent by oscillators 1, 2, 3 and 4 ( s , , , ); the second event is thereception of these pulses after τ time units ( r , , , ), which makes oscillator 5 to generateone pulse due to a supra-threshold input ( s ); the third event is the reception of this pulseat time τ ( r ); the last event is the pulse generation from oscillators 1, 2, 3, and 4 uponreaching the threshold. If the event sequence is preserved, the condition B = τ + 1 − p , follows from the periodicity of the orbit. A numerical example of this orbit structure ispresented in table V. TABLE IV. Analytic table of condition for an unperturbed S × S dynamic. event time φ , φ , φ , φ φ event num. s , , , r , , , ; s τ H (cid:15) ( τ ) = p , H (cid:15) ( B + τ ) > → r τ H (cid:15) ( p , + τ ) = p , τ s , , , τ + 1 − p , → τ + 1 − p , TABLE V. Analytic prediction of phase dynamic for parameters τ = 0 . , (cid:15) , I = 1 . and γ = 1 ,realizing a S × S cycle. event time φ , φ , φ , φ φ event num. s , , , r , , , ; s r s , , , Unperturbed S × S × S dynamic For this map, the initial conditions are such that pulses from oscillators 3 and 4 will bereceived at time τ (cid:48) after time 0. The first event is the pulse generation from oscillators 1 and2 upon reaching the threshold ( s , ); the second is the reception of pulses from oscillators3 and 4 ( r (3 , ) at time τ (cid:48) and the pulse generation from oscillator 5 ( s ) caused by this16upra-threshold input; the third event is the receive of the pulses from oscillators 1 and 2( r , ) at time τ that forces oscillators 3 and 4 to elicit a pulse ( s , ); the forth event is thereception of the pulse coming from oscillator 5 ( r ); the last event is the pulse generationfrom oscillators 1 and 2 ( s , ) upon reaching the threshold. This map implies three periodicconditions to describe a period-one attractor: D = p , + 1 − p , , E = p , + 1 − p , , and τ = 2 τ (cid:48) + 1 − p , . A example of this structure is presented in table VII. TABLE VI. Analytic table of condition for an unperturbed S × S × S dynamic. event time φ , φ φ , φ φ eventnum. s , r (3 , ; s τ (cid:48) H (cid:15) ( τ ) = p , H (cid:15) (cid:16) D + τ (cid:48) (cid:17) = p , H (cid:15) (cid:16) E + τ (cid:48) (cid:17) > → r , ; s , τ H (cid:15) (cid:16) p , + τ − τ (cid:48) (cid:17) = p , H (cid:15) (cid:16) p , + τ − τ (cid:48) (cid:17) > → H (cid:15) (cid:16) τ − τ (cid:48) (cid:17) r τ + τ (cid:48) H (cid:15) (cid:16) p , + τ (cid:48) (cid:17) = p , H (cid:15) (cid:16) p , + τ (cid:48) (cid:17) = p , p , + τ (cid:48) = p , s , τ + τ (cid:48) +1 − p , → p , + 1 − p , p , + 1 − p , TABLE VII. Analytic prediction of phase dynamic for parameters τ = 0 . , (cid:15) = 0 . , I = 1 . and γ = 1 , realizing a S × S × S cycle. event time φ , φ φ , φ φ event num. s , r (3 , ; s r , ; s , r s , C. Perturbed dynamic, return maps.
In this appendix we present three tables that show the changes to the dynamics describedin tables I, IV and VI due an incremental perturbation δ = (0 , δ , δ , δ , δ ) , where < δ <δ < δ < δ (cid:28) . In all cases without lose of generality oscillator 1 was taken as thereferential phase to define the new cycle, since this doesn’t affect the dynamics itself butonly the point of reference. The notation is the same as in B.17 A B L E V III . P e rt u r b e d S × S d y n a m i c . e v e n tt i m e φ φ φ φ φ e v e n t nu m . s , ( , ) δ δ A + δ A + δ r τ − δ H (cid:15) ( τ − δ ) = p , a H (cid:15) ( δ + τ − δ ) = p , a τ H (cid:15) ( A + δ + τ − δ ) = p , a H (cid:15) ( A + δ + τ − δ ) = p , a r ; s , τ − δ H (cid:15) ( p , a + δ − δ ) = p , b p , a + δ − δ = p , b H (cid:15) ( τ + δ − δ ) = p , b H (cid:15) ( p , a + δ − δ ) > → H (cid:15) ( p , a + δ − δ ) > → b r τ p , b + δ = p , H (cid:15) (cid:0) p , b + δ (cid:1) = p , H (cid:15) (cid:0) p , b + δ (cid:1) = p , H (cid:15) ( δ ) = p , p , r , τ − δ H (cid:15) ( p , + τ − δ ) = p , H (cid:15) ( p , + τ − δ ) = p , H (cid:15) ( p , + τ − δ ) = p , H (cid:15) ( p , + τ − δ ) = p , p , s τ − δ + − p , p , + − p , p , + − p , → p , + − p , p , + − p , s τ − δ + − p , p , + − p , → p , − p , p , + − p , p , + − p , b s τ − δ + − p , → p , − p , p , − p , p , + − p , p , + − p , A B L E I X . P e rt u r b e d S × S d y n a m i c . e v e n tt i m e φ φ φ φ φ e v e n t nu m . s , ( , , ) δ δ δ B + δ r ; s τ − δ H (cid:15) ( τ − δ ) = p , a H (cid:15) ( τ + δ − δ ) = p , a H (cid:15) ( τ + δ − δ ) = p , a τ H (cid:15) ( B + δ + τ − δ ) > → r τ − δ H (cid:15) ( p , a + δ − δ ) = p , b H (cid:15) ( p , a + δ − δ ) = p , b p , a + δ − δ = p , b H (cid:15) ( τ + δ − δ ) = p , b H (cid:15) ( δ − δ ) = p , b b r τ − δ H (cid:15) (cid:0) p , b + δ − δ (cid:1) = p , c p , b + δ − δ = p , c H (cid:15) (cid:0) p , b + δ − δ (cid:1) = p , c H (cid:15) (cid:0) p , b + δ − δ (cid:1) = p , c H (cid:15) (cid:0) p , b + δ − δ (cid:1) = p , c c r τ p , c + δ = p , H (cid:15) ( p , c + δ ) = p , H (cid:15) ( p , c + δ ) = p , H (cid:15) ( p , c + δ ) = p , H (cid:15) ( p , c + δ ) = p , r τ − δ H (cid:15) ( p , + τ − δ ) = p , H (cid:15) ( p , + τ − δ ) = p , H (cid:15) ( p , + τ − δ ) = p , H (cid:15) ( p , + τ − δ ) = p , p , + τ − δ = p , s τ − δ + − p , p , + − p , p , + − p , p , + − p , → p , + − p , s τ − δ + − p , p , + − p , p , + − p , → p , − p , p , + τ − δ + − p , = p , b b s τ − δ + − p , p , + − p , → p , − p , = p , c p , − p , p , b + p , − p , = p , c c s τ − δ + − p , → p , − p , p , − p , p , − p , p , c + p , − p , A B L E X . P e rt u r b e d S × S × S d y n a m i c . e v e n tt i m e φ φ φ φ φ e v e n t nu m . ( s , ) ; s , ( )) δ D + δ D + δ E + δ r , ; s τ (cid:48) H (cid:15) (cid:16) τ (cid:48) (cid:17) = p , H (cid:15) (cid:16) τ (cid:48) + δ (cid:17) = p , H (cid:15) (cid:16) D + τ (cid:48) + δ (cid:17) = p , H (cid:15) (cid:16) D + τ (cid:48) + δ (cid:17) = p , H (cid:15) (cid:16) E + τ (cid:48) + δ (cid:17) > → r ; s , τ − δ H (cid:15) (cid:16) p , + τ − τ (cid:48) − δ (cid:17) = p , a p , + τ − τ (cid:48) − δ = p , a H (cid:15) (cid:16) p , + τ − τ (cid:48) − δ (cid:17) > → H (cid:15) (cid:16) p , + τ − τ (cid:48) − δ (cid:17) > → H (cid:15) (cid:16) τ − τ (cid:48) − δ (cid:17) = p , a r τ p , a + δ = p , H (cid:15) ( p , a + δ ) = p , H (cid:15) ( δ ) = p , H (cid:15) ( δ ) = p , H (cid:15) ( p , a + δ ) = p , r τ − τ (cid:48) H (cid:15) (cid:16) p , + τ (cid:48) (cid:17) = p , H (cid:15) (cid:16) p , + τ (cid:48) (cid:17) = p , H (cid:15) (cid:16) p , + τ (cid:48) (cid:17) = p , p , p , + τ (cid:48) s τ + τ (cid:48) + − p , p , + − p , → p , + − p , = p , a p , p , + τ (cid:48) + − p , s τ + τ (cid:48) + − p , → p , − p , p , + − p , p , + − p , p , + τ (cid:48) + − p ,
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