Attractor metadynamics in terms of target points in slow-fast systems: adiabatic versus symmetry protected flow in a recurrent neural network

In dynamical systems with distinct time scales the time evolution in phase space may be influenced strongly by the fixed points of the fast subsystem. Orbits then typically follow these points, performing in addition rapid transitions between distinct branches on the time scale of the fast variables. As the branches guide the dynamics of a system along the manifold of former fixed points, they are considered transiently attracting states and the intermittent transitions between branches correspond to state switching within transient–state dynamics. A full characterization of the set of former fixed points, the critical manifold, tends to be difficult in high–dimensional dynamical systems such as large neural networks. Here we point out that an easily computable subset of the critical manifold, the set of target points, can be used as a reference for the investigation of high–dimensional slow–fast systems. The set of target points corresponds in this context to the adiabatic projection of a given orbit to the critical manifold. Applying our framework to a simple recurrent neural network, we find that the scaling relation of the Euclidean distance between the trajectory and its target points with the control parameter of the slow time scale allows to distinguish an adiabatic regime from a state that is effectively independent from target points.


Introduction
Coexisting fixed point attractors such as place cells [1] are commonly assumed to strongly influence cognitive processing in the brain, either alone [2] or in conjunction with feed-forward processing, with the latter being the case for the episodic memory [3]. A system characterized by a single fixed point attractor could however not be functional on its own, as it would depend on additional mechanisms to reset the dynamics. It is hence interesting, that neural activity characterized by transitions between multiple meta-stable attractors [4] has been discovered in the olfactory system of zebrafish [5] and in the gustatory cortex [6]. Similar transient state dynamics [7] is also found in resting state networks in low-frequency contributions of human fMRI data [8], where it enables processes that are associated with cognitive tasks even in the complete absence of external stimuli [9]. Resting-state brain networks also show complex spatio-temporal dynamics, in terms of transitions between states characterized by high and low functional connectivities, which resemble transiently existing attractor structures [10]. Such state-dependent fluctuations may play an important role in task-related brain computations [11] such as the interaction of motion and sensation (cf [12]).
Dynamics involving switching transitions between transiently stable states has been addressed in the contexts of semantic learning in autonomously active networks [13,14], within reservoir computing [15] and in networks dominated by heteroclinic orbits [16]. In case of the latter, periodic orbits are formed when the dynamics follows heteroclinic connections between saddle points encoding information in the different states, which however exist only for symmetry-invariant networks. The details of the internal dynamics are on the

A reference manifold on the critical manifold
Most of the stable adiabatic fixed points have in general nothing to do with the long-term behavior of the system. We will define in this context the unique set of target points as a subset of AFP corresponding to a given trajectory. We shall concentrate on trajectories that evolve on an attractor of the full system, i.e., on periodic or chaotic orbits of the full system that are reached after prolonged times. For an arbitrary locus on such a trajectory, a target point will be defined as the corresponding point that would be approached under the time evolution with fixed slow variables.
As the shape of the manifold of target points only depends on the fast subsystem and the trajectory it corresponds to, this manifold can be used as a reference manifold for the motion of the full system. Since slow variables in real world applications are not infinitesimally slow, the distance between the trajectory and the respective target points, being larger than zero, is an important measure for the assessment of the influence a target point and hence the critical manifold as a whole exerts on the dynamics. The critical manifold is both uniquely defined and numerically accessible; the manifold of target points, being a subset of the critical manifold, is also uniquely defined, as it corresponds to a certain trajectory. Therefore we will use the manifold of target points in this approach as a low dimensional reference manifold to characterize the corresponding dynamics in a potentially high dimensional phase space.

Adiabatic and non-adiabatic regimes
Analyzing the distance between a given orbit and its associated target points we discriminate two qualitatively different dynamical regimes: • In the adiabatic regime the dynamics is effectively influenced by target points, with the average distance of the trajectory to the target points scaling with the ratio of the slow and the fast time scale. We emphasize that the dynamics of the slow subsystem does not vanish in the adiabatic regime. The term relates rather to a regime of parameters, for which the trajectory follows the target points (which are adiabatic fixed points) closely, interseeded with jumps between different branches of the critical manifold. The corresponding different branches of the manifold of target points represent different transiently attracting states.
• In the non-adiabatic regime the dynamics effectively decouples from the target points, with the distance to the target points being in essence invariant with respect to changes of the relative time scale.
One may discriminate equivalently between regimes in which a perturbation expansion in the slow time scale converges or diverges, respectively [19]. Our approach aims on the other hand to develop tools suitable for the numerical study of high-dimensional slow-fast systems. We find that the study of target points, which are straightforward to determine numerically, allows in this context to distinguish between the adiabatic and the non-adiabatic regime.

Three-site recurrent neural networks
In the second part of this study we apply our framework to a network of continuous-time rate-encoding neurons, which has been shown previously to exhibit non-trivial dynamical states [36]. As we are mainly interested in how AFP influence the intrinsic behavior of the network, we restrict ourselves to autonomously active networks, i.e. networks without external input. A separation between the time scale of the membrane potential and an intrinsic neural parameter, the threshold, is present in this system. For concreteness we study a three-neuron network allowing for an in-depth understanding of the resulting dynamical states. Computing the transiently attracting states (as connected sets of target points) and the distance measures quantifying the influence of target points on the actual dynamics, leads to the characterization of two distinct dynamical regimes, i.e. the adiabatic and the non-adiabatic regime. We conclude with an analysis of the transition between these two regimes and of the role of target points for chaotic motion.

Theory-Attractor metadynamics in slow-fast dynamical systems
We consider with the slow variables. The slow time scale is set by ò b , which is also the ratio of time scales.
The set of fixed points in the fast subsystem, i.e. t x d d 0 = for given and fixed b, is of special interest for the analysis of slow-fast dynamical systems. The entirety of these points, which are the intersections of the nullclines x t d d 0 i = of the fast subsystem [37], are generally termed slow manifold [18] or critical manifold [19]. Please note that the term slow manifold is used in different contexts and fields and also with different definitions as already pointed out by Lorenz [38] (cf [18,19,21]). In the context of this work we will refer to the set of fixed points in the fast subsystem as critical manifold, as this nomenclature is widely used in the field. Here we constrain ourselves to a fast subsystem with only fixed point attractors. This is for instance the case if the fast subsystem is a gradient system, e.g. when the fast dynamics is derived from a generating functional [36,39]. The different branches of the critical manifold are therefore exclusively composed of isolated fixed points. They are often well characterized, for real-world systems, in terms of their physical and/or neurobiological properties [8].
2.1. Adiabatic fixed points (AFP) and target points In physics terminology the limit ò b →0, i.e. when the slow subsystem is infinitely slow, is termed the adiabatic limit, such as in the Born-Oppenheimer approximation [40], where the slow movement of the atomic nuclei can be treated as parametric variables when addressing the relatively fast dynamics of the electrons. In the case ò b =0 the configurationb of the slow subsystem is constant and can be treated as parameters. We will hence use the term adiabatic fixed point (AFP) for the fixed points of the fast subsystem. Note that the set of all AFP is equivalent to the critical manifold [19] of the system.  (2) is unique for a given pair (x 0 , b 0 ), depending furthermore not on ò b , which is the ratio of the time scales. All target points are AFP and a set of target points corresponding to a given trajectory is a subset of the critical manifold. This set of target points can thus be used as a reference manifold that allows to analyze the dynamics of the overall system relative to the critical manifold.
The relation of target points, trajectory and critical manifold is sketched schematically in figure 1(b). We also present in table 1 (cf appendix A) a pseudo-code for the numerical computation of AFP and target points.

Kinetic energy of phase space evolution
Target points can be computed via a straightforward evolution of the equations of motion of the fast subsystem for fixed slow variables b. Unstable adiabatic fixed points may be found [35], in addition, by minimizing the kinetic energy = + (˙˙) of the full system, for which also the flow in the slow subsystem, ḃ , needs to vanish. The same holds for slow points, which are defined by a vanishing q x and small values of q, see [35].

Attractor metadynamics
The set of target points corresponds to the mapping (2) of a trajectory t t x b , { ( ) ( )}onto the critical manifold. As mentioned before, we shall only consider trajectories on attractors for this study. The set of target points hence has the same dimensionality as the respective attractor, e.g. one dimensional for a limit cycle or fractal for a chaotic attractor (cf section 3.1). In typical slow-fast systems the embedding dimension of the critical manifold is usually much lower than the dimension of the full system. The dimension of a set of target points corresponding to a given trajectory, being a subset of the critical manifold, is thus even lower, reducing therefore the complexity of the analysis.
One defines as attractor metadynamics the time evolution of the target points equation (4), which can be continuous or characterized by jumps between distinct sets of target points. Distinct branches of the critical manifold can often be classified in real world applications, e.g. in the neurosciences when using a slow feature analysis [41]. States on the same branch of the critical manifold are then lumped together, with distinct branches corresponding to different objects, such as 'chair' and 'table'. Target points continuously connected, compare figure 1, can then be considered a set of states corresponding to the same object and hence as a transiently attracting state. The mapping (4) therefore implies that the flow moves from one transiently attracting state to another whenever the respective target point jumps from one branch of the critical manifold to the next. This classification is considered important especially in the neurosciences, where attracting states guide decision making, memory storage and recognition [4].

Distance to target points
As a measure of the influence a given target point exerts on the trajectory we will consider the Euclidean distanced(t), between the fast components of the trajectory . The corresponding time-average d á ñ has been defined in (5) for the case of a limit cycle with period τ, with a corresponding straightforward generalization for chaotic attractors.
The average distance d á ñ vanishes, per construction, in the adiabatic limit ò b →0. For finite ò b >0 we find, on the other hand, two regimes, with d á ñ scaling with ò b in the adiabatic regime, but not in the non-adiabatic regime.
Apart from the average distance d 0 á ñ  , one can also evaluate the probability density function ρ(d) of the distances over a given attractor. The corresponding cumulative distribution function of distances can then be used to characterize regimes. As a trajectory stays close to the target points for prolonged time spans in the adiabatic regime one finds contributions at small distances d 1 b   for the cumulative distribution. A second contribution comes from the jumps between the different branches of target points at larger distances d?ò b (usually d∼1). However, in the non-adiabatic case only the latter contribution, at larger distances, exists, as the system comes close to the target points only occasionally. A lack of contributions at small distances d∼ò b is an evidence that the dynamics is in the non-adiabatic regime.

Three-neuron system
To investigate the effect of target points on the overall dynamics, we consider for this study a small neural network (cf figure 2) of three rate-encoding continuous-time point neurons [36]. Networks of three neurons have already served as model systems in different contexts such as modeling pacemaker circuits [42], the stomatogastric ganglion in lobster [43] or neural motifs [44]. Here we chose the three-neuron layout to study the fundamental properties of AFP and target points, as it allows for a full investigation of its non-trivial dynamics and phase diagram.
The fast subsystem corresponds in this case to the time evolution of the membrane potential x i , with w ij >0 and w ij <0 denoting excitatory and inhibitory connection j→i respectively. The corresponding firing rate y 0, 1 i Î [ ], the neural activity, is a sigmoidal function of the respective membrane potential: We have used constant gains a i ≡a=6, whereas the threshold b i =b i (t) is adapting slowly [45] on extended time In the context of this model the ratio of times scales ò b is also called the adaption rate of the slow variables b j . The time evolution equation (9) of the slow variables attempts to drive the dynamics towards y i →1/2 as b 0  , which is the fixed point of the full system (cf appendix B).
The state of a neuron is hence described by the tuple x t b t , For the particular three-neuron system sketched in figure 2, the dynamics of the fast variables x i is given by [33] x with w 13 =w 31 <0 being inhibitory. All remaining synaptic weights are unity w 12 =w 21 =w 23 =w 32 =1 and thus excitatory connections. For the results presented here we computed the numerical solution of the ODE system equations (9) and(10) performing a fourth order Runge-Kutta integration algorithm with Fehlberg tableau [46] using step size t d 10 2 = -. For computing the AFP we used a minimization algorithm with a BFGS strategy [47,48] provided by the dlib optimization library [49] for the C++ programming language.

Symmetries
The network shown in figure 2 is symmetric under the exchange 1 3 « of the first and the third neuron, a reflection symmetry. For the special case w 13 =−1 the additional C3 rotational symmetry x is present. Equation (11) can be verified by inspecting equations (9) and (10). This leads to y y y y y y 1 , , . 12 Applying this iteration three times yields the identity transformation, equation (11) is hence equivalent to a C3 symmetry.

Dynamics of the three-neuron system
Before analyzing the transiently attracting states of the three-neuron system we briefly discuss here the possible types of long-term dynamics in the network, i.e. attractors in the phase space of the full system, for different cases of the inhibitory weightw 13 and of the time scale differenceò b . a. C3 symmetry Due to the C3 symmetry of the three-neuron system for w 13 =−1 discussed in section 3.1.1, we find a traveling wave solution where all neurons show the same activities x t x t ) with period τ, albeit with distinct phases θ i , shifted respectively by τ/6. One can easily prove that this type of motion is always an exact solution in the case of w 13 =−1 (cf appendix C).
In the top panel of figure 3 we show an example of the traveling waves solution for w 13 =−1,   full or low activity states, y i ≈1 and y i ≈0 respectively, while the second neuron stays mostly at y 2 ≈1/2 (half active). The relative phase shift is b. No C3 symmetry In figure 5 we outline the distinct dynamical regimes observed for the general case of arbitrary inhibitory weights close to the symmetric case. The relative phase shift Δ 13 is color-coded and shown as a function of the adaption rate ò b and of the inhibitory weight w 13 .
On the horizontal center axis w 13 =−1, which represents the case of a C3 symmetric system, the symmetry protected traveling wave solution with Below, we will discuss further horizontal and vertical cuts through parameter space, indicated by the dashed gray lines in figure 5, as well as the occurrence of chaotic attractors for certain parameter values to be found in the area that is green shaded in figure 5.
In figure 6 we present the sketch of the behavior along the previously mentioned two cuts of the phase diagram for ò b =10 −5 and  from anti-phase(i), via traveling waves(ii)-(iv), to in-phase oscillation(v). In the extreme cases of w 13 =−1.1 and w 13 =−0.92 we can guess once again that the firing rate is shaped by the anti-symmetric and the symmetric transiently attracting states respectively. In contrast to the case of lower adaption here we find that the transition between in the in-phase and the anti-phase oscillation of the adiabatic regime happens via the non-adiabatic regime when passing the region of symmetry protection close to the symmetric case w 13 =−1.

Transiently attracting states in the three-neuron system
In the bottom panels of figures 3 and 4 we present the firing rate y 1 of the first neuron (green) and the y 1 component of the corresponding target point (red bullets and line), with the adiabatic evolution indicated by the black arrows. , the non-adiabatic case shown in figure 3, we find that the set of target points splits into four branches. The jumps between the distinct branches are however not directly visible in the original trajectory, which is almost completely decoupled of the dynamics of the target points.  = -, a case from the adiabatic regime illustrated in figure 4, we observe on the other hand only two branches of target point manifolds. For extended time spans the trajectory follows closely the critical manifold, jumping however periodically between the two distinct branches. One observes in figure 4, that the orbit needs a certain time, a delay, to leave a given branch of the target point manifold. This is a phenomenon typical for systems with multiple time scales [28].

Stable and unstable adiabatic fixed points
In figure 7, which illustrates the dynamics along the horizontal cut through the phase diagram for w 13 =−1, the adiabatic fixed points for five different values of the adaption rate are shown. Here we have included all adiabatic fixed points, including the unstable ones, which we have found by minimizing equation (3). Each of the five insets shows a sketch of the trajectory (green), target points (red), other stable AFP (blue) and saddle AFP (gray) in the activity y 1 of the first neuron (cf bottom panels of figures 3, 4). Each of these insets serves as an example of the five different phases that can be distinguished by the shape and stability of the AFP. The whole non-adiabatic regime with phase shift 1 3 13 D = is marked by the yellow bar, while the adiabatic regime with 1 2 13 D = is indicated by a brown bar. The range of occurrence in the adaption rate ò b of the four non-adiabatic phases is indicated by the gray bars included in the yellow bar.
Starting at relatively high adaption ò b =10 −2 in figure 7 panel(v) we find only stable AFP of which all act as target points and the attractor metadynamics therefore is continuous. Decreasing the adaption rate ò b (cf panels (iv) to (i)) we observe the occurrence of saddle type AFP and stable AFP which are no targets to the trajectory.
In panels (i)-(iii) of figure 7 one can find saddle-node bifurcations, in which stable (blue) and saddle type (gray) AFP merge. The saddle-node bifurcation points originate from a cusp bifurcation [51] on the critical manifold, with the slow variables as bifurcation parameters. The adaption rateò b determines at which point the trajectory crosses the cusp bifurcation. Thus one either finds a continuous manifold of target points or jumps.  =´the firing rates change in a smooth manner from anti-phase oscillation to traveling waves, and with a small jump to in-phase behavior.
This change in the structure of the corresponding target points with the adaption rate also affects the attractor metadynamics, which becomes discontinuous as well.
• For all adaption rates we find stable traveling waves solutions with a relative phase difference 1 3 13

D =
, for which the trajectory is only marginally influenced by the transiently attracting states. This is the non-adiabatic regime (yellow bar).
• For lower adaption rates Here, in the adiabatic regime (brown bar), the motion follows strictly the metadynamics of the target points with some delay at the jumps, as mentioned earlier.
We emphasize that the transition between the adiabatic and non-adiabatic regime occurs at a remarkably low value of the adaption rate  > -· is hence strongly protected by the C3 symmetry of the system. Off symmetry, i.e. for w 1 13 ¹ -, the transition between the adiabatic and the non-adiabatic regime may become continuous (compare figure 5), and shifts to higher values in ò b . Along the symmetric axis w 13 =−1 we find a small region in the adaption rateò b close to the transition, as marked by stripes in figure 7, where attractors from both the adiabatic and the non-adiabatic regime coexist. Starting close to an attractor from the non-adiabatic regime we can trace the non-adiabatic regime decreasing the adaption rateò b in small steps. This procedure succeeds up to a certain value of ò b , depending on the precision of the computation and the step size in ò b , before ending up in an attractor of the adiabatic regime. Vice versa we can start from the adiabatic regime and trace it increasing ò b . This means we can observe hysteresis in the bistable transition region between the regimes. Figure 8 shows the average distance d á ñ, as defined by equation (5) An adiabatic regime is always observed for low adaption rates ò b . It is however clearly evident from figure 8 that the adiabatic regime is pushed down for the symmetric case w 13 =−1, by more than one order of

D =
). There are five phases to be distinguished by the stability and the shape of the AFP. Each phase is represented by a characteristic trajectory-AFP-plot in y 1 (cf figure 3 and 4 bottom row), the range of occurrence for the middle three is indicated by the gray bars within the yellow bar. Green/red/blue/gray in the plots denote the trajectory/target points/other stable AFP/unstable AFP respectively. For relatively large values of the adaption rateò b (v) there are only stable AFP, acting as the target points. Decreasing ò b saddle AFP and non-target stable AFP occur (i-iv). Hysteresis in the transition is observed in the striped area. magnitude. States protected by symmetry operations, in our case the traveling wave solution illustrated in figure 3, can be exceedingly stable.
The discontinuous transition between the two regimes in the symmetric case w 13 =−1 is confirmed by both figure 5 and figure 8. There is a transition region in ò b , where two arbitrarily close initial conditions can show fundamentally different dynamics being in the adiabatic and the non-adiabatic regime, respectively (cf figure 7 and the related discussion). On the two sides of this transition region, only one of these two different kinds of dynamics exists. Lacking any intermediate kind of dynamics, this abrupt transition is remarkable since the shape of the AFP manifold does not depend on the adaption rate ò b at all. We thus stress that the distance to the reference manifold of target points, which is unique for given parameter values and initial conditions, reveals a qualitative change in the dynamics.

Statistics of the distance between trajectory and target points
For the same parameters as used in figures 3 and 4 we present in figure 9 the cumulative distribution function P(d) of the distance d, as defined by equation (6), between the trajectory and the corresponding target points. First the relative density ρ(d) of distances averaged over a given attractor is computed numerically by sampling the measured distances to 400 bins on the logarithmic range d 10 , 2 To obtain the cumulative distribution of distances the density is then integrated as defined by equation (6).  (5). Shown are horizontal cuts through the phase diagram, figure 5, along w 13 =−1.1 (gray), −1 (yellow) and −0.92 (brown). The regime is non-adiabatic, when the distance becomes independent of ò b , as it happens for larger values of the adaption rate. The system is adiabatic, conversely, when the distance scales with ò b , which happens for the symmetric casew 13 =−1 only for low adaption rates. ] close to unity, resulting in turn from small residual variations of distance. For the adiabatic case (cyan) we observe substantial contributions both close to unity, reflecting the jumps between different target branches, and extended contributions from small distances d<10 −2 , which result from the evolution close to the AFP manifold. The difference between the adiabatic and the non-adiabatic regime shows up prominently in the statistics of the distances between trajectory and target points.  < -· and near the symmetric case, w 1, 0.93 13 Î --[ ] , we find patches in the parameter space that exhibit chaotic motion. An example is shown in figure 10 for ò b =10 −5 and w 13 =−0.970 9. The trajectory and the corresponding target points are shown both as a time series (left panel) and projected to the y 2 −y 1 plane in phase space. Applying a recently developed 0−1 test for chaos, based on the cross-distance scaling of initially nearby trajectories [52], we find that this attractor is indeed chaotic.

Target points corresponding to chaotic motion
Except for some overshooting, the trajectory shows a similar behavior to the adiabatic motion presented in figure 4, approaching the target point manifold and staying then close to it as long as it remains stable.
The manifold of target points has a highly non-trivial structure for the chaotic motion, in contrast to the piece-wise smooth and periodic structure observed for the case of limit cycle attractors, as observed e.g. in figure 7. A visual inspection indicates (see the inset in the right-hand panel of figure 10) that the phase space projection of the target points forms a fractal structure. We did not attempt to directly compute the fractal dimension of the manifold of target points shown in figure 10, as this is computationally highly demanding. Chaotic behavior, on the other hand, is typically linked to the presence of fractals in the dynamical behavior [53].
Speaking in terms of transiently attracting states this means that in case of chaotic motion we cannot describe the motion as switching between two transiently attracting states. But rather we find that there must be infinitely many branches of transiently attracting states forming a fractal set. One may nevertheless cluster these fractal sets into two broad classes, corresponding to small and large y 2 respectively. A possible interpretation of that behavior includes a chaotic motion of the overall system driving the dynamics along the critical manifold, which itself does not have a fractal structure. Thus the irregular motion near the critical manifold results in a fractal set of target points. We emphasize that the target points themselves are always stable fixed points of the fast subsystem, for any values of the slow variables.
In order to compare the chaotic motion more precisely to the adiabatic and the non-adiabatic regimes, we have included in figure 9 the cumulative distribution function of distances (light green line) for the chaotic attractor. Besides the contribution close to unity due to the large distance jumps between different AFP branches, it reveals contributions at small distances d<10 −2 , which are significant also for the adiabatic case (cf to cyan line). In this respect the chaotic attractor is close to the adiabatic regime. We find however additional medium size contributions d≈0.3 that result from smaller jumps within the fractal set of target branches.
The chaotic dynamics thus goes through an infinite series of transient states, trying to stay close to the critical manifold (itself having a simple, not fractal geometry), which is also confirmed by the distribution of distances to the target points. We do not observe chaotic exploration of the phase space, i.e. long detours leading away from the target points as one could imagine for chaotic dynamics. Due to the relatively small adaption rate the chaotic Figure 10. Chaotic attractor (green) of the three-neuron system for ò b =10 −5 , w 13 =−0.970 9 (indicated by the red framed triangle in figure 5) with the corresponding target points (red). Shown are the firing rate y 1 of the first neuron over time (left) and the projection of the orbit to the y 1 −y 2 plane (right). The activity of the neuron is attracted towards fragments of the target point manifold, performing smaller and larger jumps between these fragments. The inset in the right-hand panel corresponds to a zoom-in to a region of target points, which on visual inspection seem to form a fractal structure. dynamics is guided along different branches of the target manifold-similar to the regular adiabatic motion presented in figure 4.
Keeping on the other hand in mind that the inhibitory weight w 0.970 9 13 =is only slightly off the symmetric case, it is to be expected that the shape of the critical manifold only changes smoothly in w 13 . Therefore we would expect the same qualitative properties for the dynamics, i.e. the trajectory moving mostly close to branches of the target point manifold, as in the symmetric case presented in figure 4. But the origin of the chaotic dynamics is thus not obvious.

Discussion and conclusions
We have proposed here that the study of adiabatic fixed points and transiently attracting states, which are sets of target points, are useful when trying to understand complex slow-fast systems. As these can be realized by adding an additional slow component to an attractor network, the target points have a well defined physical function representing, e.g., as cognitive states in a neural network. Both the location in phase space and the topology of the manifold of target points can be easily evaluated without explicit knowledge of the full set of fixed points in the fast subsystem. The mapping of a given trajectory onto the corresponding set of target points is, by definition, unique and does not depend on the time evolution on the slow time scale. Thus target points qualify as a unique reference manifold for the overall dynamics. Transitions between smooth subsections of the manifold of target points correspond generically to transitions between well defined biological, physical or cognitive states.
We have shown, analyzing a three-site network of adapting rate-encoding neurons, that system symmetries may stabilize peculiar solutions which effectively decouple from the dynamics of target points. This decoupling shows up also in the statistics of the Euclidean distance between the trajectory and the respective target points. The distance distribution can be used furthermore to classify states in terms of the relevance of the reference manifold.
We applied our analysis both to limit cycle and to chaotic dynamics. The detailed analysis of the latter phenomenon is a possible subject of future work. We find, somewhat surprisingly, that a fractal set of target points may guide chaotic behavior.
The above-described phenomenon is not to be confused with the case when the attractor of the fast subsystem is more complicated than a fixed point, e.g. limit cycles or chaotic attractors. In future studies the above described method for distinguishing dynamical regimes by means of target (fixed) points could be generalized to arbitrary target attractors. The key implication then will be to generalize the distance measurement between a trajectory and the corresponding attractor.
Our approach assumes a separation of time scales to be present, breaking down once the time scales for the slow and for the fast subsystem approach each other. The ability to treat the system analytically is however not needed.
We believe, in conclusion, that the study of target points could provide additional insights especially for slow-fast dynamical systems with a relatively high number of degrees of freedom. These points can serve as a unique reference to investigate the overall dynamics of a system and are easily computable. It would be interesting to follow the possibility to extend the here proposed approach to non-autonomous systems, such as the modulated neural networks processing cognitive stimuli.

D =
t between the first and the third neuron, where τ is the period of the solution. We make an ansatz for the solution with two periodic functions x(t), b(t) of period τ and an arbitrary shift θ. The corresponding firing rates of the neurons therefore are given by y t y t y t y t y t y t , , , where the notation for the periodic function y t y x t a b t , , = ( ) ( ( ) ( )) is used. From this we get to the resulting equations of motion for the membrane potential x t x t y t y t x t x t y t y t x t x t y t y t ORCID iDs Hendrik Wernecke https://orcid.org/0000-0002-0187-3079 Bulcsú Sándor https:/ /orcid.org/0000-0003-4887-7859 Claudius Gros https:/ /orcid.org/0000-0002-2126-0843 Table 1. Short pseudo-code description how to compute the adiabatic fixed points (AFP) and target points.