Paper deep dive
Analyzing the Dynamics of WSBM-based Echo State Networks using InputDSA
Huy Mai, Robert Clarke
Intelligence
Status: not_run | Model: - | Prompt: - | Confidence: 0%
Entities (0)
Relation Signals (0)
No relation signals yet.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Bridging the structural gap between biological brains and artificial neural networks presents a pathway to altered computational dynamics. This paper introduces a Reservoir Computing framework that replaces random networks with biologically realistic modularity by applying the Weighted Stochastic Block Model (WSBM) to Echo State Networks (ESNs). This topological parameterization constructs reservoirs reflecting latent functional sub-circuits. Furthermore, the integration of Input Dynamical Similarity Analysis (InputDSA) provides a mechanism to isolate and compare how structural priors modulate intrinsic recurrent dynamics versus external inputs. By evaluating these architectures, we establish that structural diversity does not necessarily predicts behavioral diversity across the selected network subset. Furthermore, our demixed analysis confirms Anna Karenina principle of dynamics, demonstrating that environmental constraints enforce dynamical convergence in successful agents, while failing networks degenerate into idiosyncratic trajectories.
Tags
Links
- Source: https://arxiv.org/abs/2610.15380v1
- Canonical: https://arxiv.org/abs/2610.15380v1
Trouble viewing inline? Open PDF directly →
Full Text
38,647 characters extracted from source content.
Expand or collapse full text
1–18, 2026 Analyzing the Dynamics of WSBM-based Echo State Networks using InputDSA Huy Maiminhhuymaiduc@gmail.com Vietnam National University Ho Chi Minh City, University of Science Robert Clarkerc2205@bath.ac.uk University of Bath Abstract Bridging the structural gap between biological brains and artificial neural networks presents a pathway to altered computational dynamics. This paper introduces a Reservoir Com- puting framework that replaces random networks with biologically realistic modularity by applying the Weighted Stochastic Block Model (WSBM) to Echo State Networks (ESNs). This topological parameterization constructs reservoirs reflecting latent functional sub- circuits. Furthermore, the integration of Input Dynamical Similarity Analysis (InputDSA) provides a mechanism to isolate and compare how structural priors modulate intrinsic re- current dynamics versus external inputs. By evaluating these architectures, we establish that structural diversity doesn’t necessarily predicts behavioral diversity across the selected network subset. Furthermore, our demixed analysis confirms Anna Karenina principle of dynamics, demonstrating that environmental constraints enforce dynamical convergence in successful agents, while failing networks degenerate into idiosyncratic trajectories. Keywords: Weighted Stochastic Block Model, Echo State Network, Dynamical Similarity Analysis 1. Introduction Reservoir computing (RC), typically implemented as echo state networks (ESNs), provides an efficient approach to processing temporal data by leveraging the nonlinear dynamics of a fixed recurrent hidden layer. This architecture avoids the computational overhead of gradient-based optimization through time. However, standard ESNs rely on unstructured, random graphs (e.g., Erd ̋os–R ́enyi) (Gauthier et al.), diverging from biological neural net- works. Biological brains utilize organized meso-scale community structures to optimize in- formation routing and processing efficiency (Betzel et al.). By omitting these connectomic priors, artificial reservoirs fail to exploit the structural advantages of biological topologies. To integrate biological plausibility into RC, we propose a connectome-constrained frame- work using the Weighted Stochastic Block Model (WSBM) (Aicher et al., b,a). The WSBM extracts latent block structures and weight distributions from real-world connectome data, enabling the construction of ESN reservoirs with meso-scale modularity. This prior replaces random initialization, imposing biological constraints on the dynamical system. Evaluating these priors requires analyzing the network’s internal temporal mechanisms rather than static spatial geometries. Because external inputs continuously perturb the reservoir’s state, structural interpretation requires isolating this external forcing from in- trinsic recurrence. We utilize Input Dynamical Similarity Analysis (InputDSA) (Huang © 2026 H. Mai & R. Clarke. arXiv:2610.15380v1 [cs.AI] 11 Oct 2026 Mai Clarke et al.), which extends standard DSA (Ostrow et al.) via Subspace Dynamic Mode Decom- position with control (SubspaceDMDc). This method demixes intrinsic network dynamics from input-driven dynamics. By comparing the dynamical manifolds of random reser- voirs against WSBM-ESNs, InputDSA provides a quantitative basis to demonstrate how connectome-derived topologies alter the temporal structure of computation. The primary contributions of this paper are: • We introduce a generative framework that parameterizes ESN reservoirs using WSBM, embedding biological meso-scale modularity into the network topology. • We apply InputDSA to isolate input-driven representations from intrinsic recurrent dynamics, providing qualitative evidence of how structural priors shape internal tem- poral flow 2. Related works A brief introduction to SBMs and ESNs is provided in Appendix A. Reservoir Computing.Reservoir Computing (RC) emerged as a unifying paradigm (Verstraeten et al., 2005, 2007) that consolidated models such as Echo State Networks (ESNs) and Liquid State Machines (LSMs) to address the training bottlenecks of standard Recurrent Neural Networks (RNNs). By decoupling the recurrent hidden layer—the reser- voir—from the readout layer, temporal input signals are projected into a high-dimensional, non-linear state space through an untrained recurrent substrate. Because the reservoir re- mains fixed and only the linear readout requires training, RC is versatile; the recurrent substrate can be instantiated not only as artificial nodes in software, but also across diverse physical mediums, including photonic (Vandoorne et al., 2008), electronic (Kulkarni and Teuscher, 2012), and mechanical systems (Hauser et al., 2011). While traditionally rely- ing on uniform or Gaussian random graph models, recent work demonstrates that careful architectural design of the recurrent substrate yields quantifiable improvements. For in- stance, Gallicchio and Micheli showed that constrained reservoir topologies and structured organization of recurrent units can provide notable gains in predictive performance. Connectome-Constrained Networks.One way to constrain reservoir creation is through biologically grounded information. McDaniel et al. demonstrated that reservoirs endowed with biologically-inspired hierarchical structures achieve distinct performance char- acteristics compared to standard random graphs, and recent work by Milisav et al. (2026) further established that hierarchical modularity in neuromorphic reservoirs enhances mem- ory capacity and supports multitasking. Efforts to integrate biological constraints into artificial networks generally fall into two categories: direct instantiation of empirical con- nectome weights (Shi et al., 2022; Lappalainen et al., 2024), and parameterized generative models that capture meso-scale topological rules (Betzel and Bassett, 2017; Akarca et al., 2021; Salova and Kov ́acs, 2025). Our work aligns with the latter, seeking to incorporate the organizing principles of biological networks. Weighted Stochastic Block Models.To capture these meso-scale biological rules, the Weighted Stochastic Block Model (WSBM) offers a robust generative framework. In network neuroscience, Faskowitz et al. demonstrated that WSBMs characterize the pat- terned interactions and community structures inherent in the human connectome across 2 WSBM-based ESN Comparison Read-in Read-out Community 1 Community 2 Community 3 Community 4 Reservoir AssortativeDisassortative Core-peripheryMixed Figure 1: Overview of the WSBM-based Echo State Network framework, with 4 main block models. the lifespan. In the context of artificial computation, complex topological features resem- bling SBM structures have been shown to be computationally advantageous. d’Andrea et al. found that the modularity and link density of biological connectomes shape the learn- ing performance of bio-inspired recurrent neural networks. Furthermore, Dehghani showed that when reservoir architectures are placed under evolutionary selection for spatiotemporal prediction, the evolved networks naturally conserve an SBM-like spectral envelope and lock macroscopic modularity to specific functional bands. Despite the established utility of WS- BMs in describing brain networks and the recognized benefits of structural priors in ESNs, existing literature has not utilized the WSBM as a direct generative prior to jointly sample both the discrete topology and the continuous weight magnitudes of artificial reservoirs. That’s the gap our paper aims to address. Representational and Dynamical System Analysis.While downstream per- formance metrics validate the utility of WSBM-generated reservoirs, they cannot justify how joint topological and weight priors alter internal information processing. To under- stand the computational mechanics of recurrent networks, it is necessary to examine their state-space trajectories. Historically, the geometric family of metrics—such as Represen- tational Similarity Analysis (RSA) (Kriegeskorte et al., 2008), Centered Kernel Alignment (CKA) (Cristianini et al., 2002; Kornblith et al., 2019), and Canonical Correlation Analysis (CCA) (Hotelling, 1936)—have been used to determine whether different neural systems represent stimuli with similar spatial arrangements in their latent spaces. However, be- cause these methods operate primarily on static or time-averaged activity, they cannot distinguish between systems that possess identical representational geometry but execute different temporal computations. To address this limitation, dynamics-focused frameworks such as Dynamical Similarity Analysis (DSA) (Ostrow et al.) evaluate networks by em- 3 Mai Clarke bedding their temporal trajectories into linear dynamical operators (via Koopman operator theory) and aligning these dynamics matrices rather than static state-space geometries. Yet, standard DSA does not account for how input drive affects these trajectories, meaning it can conflate intrinsic network behavior with external forcing. Because our WSBM-ESNs are deployed in Reinforcement Learning environments—functioning as non-autonomous sys- tems driven by environmental observations—we employ the recently proposed InputDSA framework (Huang et al.) instead. 3. Weighted Stochastic Block Model-based Echo State Network Generative model.We instantiate the exponential family of Section A.2 as a normal distribution, so that the natural parameters η z i ,z j correspond to a block mean and variance, and we retain the Bernoulli edge-existence term Θ of the binary SBM. A reservoir is then sampled from three symmetric K× K block matrices: the mean edge weight μ, the weight dispersion σ, and the edge probability Θ, W ij ∼N μ z i ,z j , σ 2 z i ,z j · Bernoulli(θ z i ,z j ), W i = 0,(1) with each undirected edge drawn once and mirrored so that W = W ⊤ , matching the undirected connectomes from which the model is estimated (Betzel et al.). The sampled matrix is rescaled to a fixed spectral radius to enforce the echo state property. This joint parameterization is the point of departure from prior work: sampling topology and weight magnitude from a single generative prior, rather than imposing empirical weights directly (Shi et al., 2022; Lappalainen et al., 2024) or generating topology alone (Betzel and Bassett, 2017; Akarca et al., 2021). Community interaction motifs.Following Betzel et al., a pair of communi- ties r,s is classified by the ordering of three block means alone: it is assortative if min(μ r ,μ s ) > μ rs , disassortative if μ rs > max(μ r ,μ s ), and core–periphery otherwise. Because μ, σ and Θ are independent, the model expresses architectures in which topol- ogy and weight structure carry different motifs. Two consequences follow. First, a scalar edge probability renders every architecture topologically identical, confining all meso-scale structure to the weights; this degenerate case is what a single connection probability p ER recovers. Second, the classification above reads means only and is therefore blind to σ, so dispersion constitutes a structural axis that the published taxonomy cannot resolve. Design space.The weight matrices admit one gauge freedom, since scaling μ and σ by a common factor leaves the reservoir invariant after spectral rescaling. The design is accordingly parameterized by five structural axes: the signed weight contrast c = (μ in − μ out )/2 expressed in units of ̄σ, the weight centre m = (μ in + μ out )/2, the dispersion ratio σ in /σ out , the topology ratio ρ = θ in /θ out , and the heterogeneity of community sizes. A signed c subsumes the assortative and disassortative labels as opposite half-axes, with c = 0 recovering the Erd ̋os–R ́enyi null. When sweeping ρ, block probabilities are solved so that the expected edge count is held fixed, ensuring a topological motif is never confounded with edge abundance. 4 WSBM-based ESN Comparison 1. Sample design space 2. Describe 8 graph descriptors 3. Stratify, MaxMin 5× 20 = 100 4. Train RL, fixed hyperparams 5. InputDSA A, B distances selection | outcome structure onlydynamics only Figure 2: Analysis pipeline. The dashed barrier marks the independence of selection from outcome: reservoirs are chosen on structural descriptors alone, and dynamical similarity is only ever a measured quantity, never a selection criterion. 4. Experiment Selection of a representative subset.Reservoirs are selected on structural descrip- tors rather than on dynamical similarity, so that selection and outcome remain statistically independent; choosing systems for dynamical dissimilarity and subsequently reporting dy- namical dissimilarity would be circular. Each sampled reservoir is characterized by eight descriptors computed from W : weighted modularity of the planted partition, mean par- ticipation coefficient, weighted clustering coefficient, global efficiency, the coefficients of variation of node strength and of betweenness centrality, the spectral ratio |λ 2 /λ 1 |, and the negative-weight fraction. Descriptors are standardized and projected by principal compo- nent analysis. The population is stratified into five classes — assortative, disassortative, core–periphery, mixed, and the Erd ̋os–R ́enyi null — and within each stratum n = 20 reservoirs are chosen by MaxMin selection in descriptor space, yielding M = 100. Stratification preserves the in- terpretability of the published taxonomy, while MaxMin selection within a stratum ensures the retained reservoirs span the within-class variation rather than clustering at its mode. The null stratum is a fifth arm of the design rather than a control appended after the fact, and it carries the burden of the central claim. Protocol.Each selected reservoir serves as the state encoder of a continuous control agent in a partially observable locomotion environment, with a model-free policy trained on the reservoir activations. All non-structural quantities — N x , ρ, α, γ, readout regularization and policy hyperparameters — are held constant across every reservoir, so that connectivity is the sole manipulated variable. RL tasks and codes are adapted from L ́eger et al. (2024) 1 under MIT license and only the developmental reinforcement loop is run; no evolutionary optimization of reservoir hyperparameters is performed. Hypotheses.The study tests two claims: H1 The Anna Karenina principle of dynamics. Task constraints enforce dynami- cal convergence in top-performing agents, while failure modes promote idiosyncratic divergence. 1. https://github.com/corentinlger/ER-MRL 5 Mai Clarke H2 Structural diversity predicts behavioural diversity. The descriptor-space spread of the selected subset predicts the spread of returns attained across it. 5. Dynamics Analysis by InputDSA A) B) Figure 3: Performance and dynamical representation of structured reservoir networks. A) Mean episode reward plot for 100 selected reservoirs compared to a Vanilla PPO baseline across three RL environments. B) Pairwise structural distance versus dynamical state distance for the top and bottom 10 reservoirs. We compare dynamics by fitting the non-autonomous state-space model V (t + δt) = AV (t) + Bs(t) using standard DMDc. Systems are then compared through the Procrustes distance between their controllability matrices, which decomposes into an intrinsic term, DSA state , and an input-routing term, DSA input (Huang et al.). Here, s(t) and V (t) denote the system input and hidden state at time t, respectively. In our experiments, s(t) concate- nates the current environment observation o t , previous action a t−1 , and previous reward r t−1 , while V (t) is the context vector c t representing the reservoir state after processing s(t), as illustrated in Fig. 9. Hypothesis 1 Our first hypothesis is built upon the Anna Karenina principle of dynamics: successful complex systems share common traits, while failures occur in unaligned ways. In reinforce- ment learning environments, achieving high rewards requires a specific set of computational responses. Consequently, top-performing networks are forced into a shared, optimal pat- tern of behavior to solve the task. Conversely, failing networks are unconstrained by the 6 WSBM-based ESN Comparison Figure 4: InputDSA distance matrices comparing the top 10 (T) and bottom 10 (B) per- forming reservoir architectures for the Swimmer-v4 task. The pairwise dynami- cal distances are computed using five metric configurations: Joint, State (Joint), Control (Joint), State (Separate), and Control (Separate). demands of success and diverge into scattered, uncoordinated trajectories. We hypothesize that successful models will exhibit aligned input-driven dynamics, whereas poor-performing models will display mismatched dynamics. We present the performance of the 100 selected reservoirs in Fig. 3A, which demon- strates significant fluctuations in mean episodic rewards for the Hopper-v4, Walker2d-v4, and Swimmer-v4 environments. This variability in reward is mirrored by a high variance in the corresponding network dynamics. However, when comparing this to the performance on the Ant-v4 and HalfCheetah-v4 tasks (Appendix B, Fig. 7), the latter exhibit substan- tially more stable learning curves. This discrepancy implies that the specific environmental constraints and degrees of partial observability inherent to certain control tasks distinctly shape the variance of the resulting dynamics. To systematically assess this, we investigated the potential correlation between struc- tural distance and the hidden state dynamics, quantified by DSA state . As illustrated in Fig. 3B, fitting a linear model reveals no significant correlation between the structural de- scriptors and the intrinsic dynamical state representations. However, a critical observation emerges from the input-driven dynamics: the structural variance among the most successful models is high, yet their dynamical dissimilarity is low. This suggests that the environments impose strict computational constraints that enforce dynamical convergence regardless of the underlying structural topology. This convergence aligns with recent findings utilizing InputDSA on control tasks like olfactory plume tracking (see Section 4. Applications Huang et al.). In these settings, top-performing agents develop similar input-driven dynamics because success depends on coordinated responses to environmental stimuli. Furthermore, analyzing the singular value spectrum of the input-mapping operator reveals that inputs successfully excite a broader range of dimensions within the state space of top agents, whereas bottom-performing agents diverge toward heterogeneous dynamics. Thus, the shared demands of the task force high performance WSBM-ESNs to converge onto an optimal, input-responsive dynamical mani- fold. 7 Mai Clarke While recent literature suggests SubspaceDMDc (via N4SID) is superior for partial ob- servable systems due to its handling of unobserved state biases, we chose to utilize standard DMDc for our WSBM-ESN analysis. When applying N4SID to the high-dimensional, cou- pled internal states of our reservoirs (Fig 8), the algorithm exhibited numerical instability. Because ESNs operate with recurrent matrices scaled near the edge of chaos, the subspace projection steps seemingly produced near-singular matrices, causing the Joint metric dis- tances to exceed 10 6 . In contrast, as featured from Fig 4, standard DMDc provided stable, bounded alignments suitable for our observable reservoir states. In general, we observed patterns of similarity across the five InputDSA matrices that val- idate our hypothesis. On the Joint metric, the low pairwise distances among the top models indicate a strong overall alignment of the non-autonomous systems. On the State (Joint and Separate) metrics: This tight alignment reveals that successful WSBM-constrained reservoirs converge onto an optimal intrinsic dynamical manifold to solve the task. On the Control (Joint and Separate) metrics: The high variance and scattered distances among the bottom models highlight that bottom agents suffer from chaotic, uncoordinated input- routing, failing to integrate environmental observations. In conclusion, these metrics confirm the Anna Karenina principle of dynamics hypothesis. Hypothesis 2 Our second hypothesis concerns whether the structural diversity engineered into the se- lected subset propagates into behavioural diversity. MaxMin selection maximizes descriptor- space spread within each stratum, so if the eight descriptors capture the axes of connectivity that govern computation, reservoirs lying far apart in descriptor space should attain mea- surably different returns. We therefore test the pairwise form of the claim: whether the descriptor distance between two reservoirs predicts the difference in the returns they achieve. Behavioural diversity is substantial. Evaluating all M = 100 reservoirs with deterministic rollouts, returns span 221–3010 on Hopper-v4, 124–1891 on Walker2d-v4, and 17.8-128.7 on Swimmer-v4, with a coefficient of variation strikingly stable across the three environments (CV = 0.54, 0.50, 0.52). The selected reservoirs are therefore far from behaviourally inter- changeable, and the environments differ in the scale of return they support rather than in the relative spread they induce. That diversity is not, however, organized by position in descriptor space. Across all 4950 reservoir pairs, the rank correlation between descriptor distance and absolute return difference is indistinguishable from zero in every environment (Spearman |ρ| ≤ 0.012, all p ≥ 0.38). Restricting to within-stratum pairs, which removes between-class variation, yields no coherent pattern either: two of fifteen tests reach nominal significance, in incon- sistent directions across environments. We read this as a statement about the descriptor basis rather than about structure itself. Connectivity evidently matters for the returns attained, since the reservoirs differ by more than an order of magnitude, but the eight de- scriptors of Section 4 do not span the axis along which it matters. A further limitation is that each reservoir contributes a single training run, so the observed spread confounds the structural contribution with the stochasticity of policy optimization; separating the two requires replicate seeds per reservoir, which we identify as the primary extension of this analysis. 8 WSBM-based ESN Comparison 6. Conclusion and Discussion This study presents a generative framework that parameterizes Echo State Networks (ESNs) using the Weighted Stochastic Block Model (WSBM) to embed biologically realistic meso- scale modularity. By applying Input Dynamical Similarity Analysis (InputDSA), we es- tablished that environmental task constraints enforce dynamical convergence in successful networks, whereas failing models diverge into uncoordinated trajectories. We have relegated the plots of the Ant-v4 and HalfCheetah-v4 tasks to Appendix B. The mean episodic reward plots for these specific environments are stable and exhibit low variance. Consequently, the dynamics across all generated systems converge in a similar manner under these specific environmental constraints. Examining these stable examples provides a clear baseline, and the mechanisms driving this uniform convergence warrant further investigation. The choice of dynamical decomposition algorithm presents a methodological limitation. Although N4SID is theoretically suited for partially observable systems, it exhibited numer- ical instability when applied to the high-dimensional internal states of our reservoirs. As a result, standard DMDc was utilized to ensure stable, bounded alignments. Future research should investigate adaptations to N4SID or alternative subspace methods that can reliably handle the specific dynamics of ESNs Finally, future work should refine the information routing mechanism of the read-in layer. Rather than employing standard random routing, directing external inputs to specific structural modules within the reservoir would allow for a more precise evaluation of how localized, meso-scale topologies process targeted stimuli. References Christopher Aicher, Abigail Z. Jacobs, and Aaron Clauset. Adapting the Stochastic Block Model to Edge-Weighted Networks, a. URL http://arxiv.org/abs/1305.5782. Christopher Aicher, Abigail Z. Jacobs, and Aaron Clauset. Learning Latent Block Structure in Weighted Networks. 3(2):221–248, b. ISSN 2051-1310, 2051-1329. doi: 10.1093/ comnet/cnu026. URL http://arxiv.org/abs/1404.0431. Danyal Akarca, Petra E. V ́ertes, Edward T. Bullmore, Kate Baker, Susan E. Gather- cole, Joni Holmes, Rogier A. Kievit, Tom Manly, Joe Bathelt, Marc Bennett, Gi- acomo Bignardi, Sarah Bishop, Erica Bottacin, Lara Bridge, Diandra Brkic, Annie Bryant, Sally Butterfield, Elizabeth M. Byrne, Gemma Crickmore, Edwin S. Dalmai- jer, F ́anchea Daly, Tina Emery, Laura Forde, Grace Franckel, Delia Fuhrmann, An- drew Gadie, Sara Gharooni, Jacalyn Guy, Erin Hawkins, Agnieszka Jaroslawska, Sara Joeghan, Amy Johnson, Jonathan Jones, Silvana Mareva, Elise Ng-Cordell, Sinead O’Brien, Cliodhna O’Leary, Joseph P. Rennie, Ivan Simpson-Kent, Roma Siugzdaite, Tess A. Smith, Stephani Uh, Maria Vedechkina, Francesca Woolgar, Natalia Zdorovtsova, Mengya Zhang, Duncan E. Astle, and the CALM team. A generative network model of neurodevelopmental diversity in structural brain organization. Nature Communica- tions, 12(1):4216, Jul 2021. ISSN 2041-1723. doi: 10.1038/s41467-021-24430-z. URL https://doi.org/10.1038/s41467-021-24430-z. 9 Mai Clarke Richard F Betzel and Danielle S Bassett. Generative models for network neuroscience: prospects and promise. Journal of the Royal Society, Interface, 14(136), 2017. ISSN 1742-5662. doi: 10.1098/rsif.2017.0623. Richard F. Betzel, John D. Medaglia, and Danielle S. Bassett. Diversity of meso-scale archi- tecture in human and non-human connectomes. 9(1):346. ISSN 2041-1723. doi: 10.1038/ s41467-017-02681-z. URL https://w.nature.com/articles/s41467-017-02681-z. Eric W. Bridgeford, Alexander R. Loftus, and Joshua T. Vogelstein. Statistical Models of Random Networks, page 98–188. Cambridge University Press, 2025. Nello Cristianini, John Shawe-Taylor, Andre Elisseeff, and Jaz Kandola. On kernel-target alignment. In Thomas G. Dietterich, Suzanna Becker, and Zoubin Ghahramani, editors, Advances in Neural Information Processing Systems 14: Proceedings of the 2001 Confer- ence. The MIT Press, 11 2002. ISBN 9780262271738. doi: 10.7551/mitpress/1120.003. 0052. URL https://doi.org/10.7551/mitpress/1120.003.0052. Valeria d’Andrea, Michele Puppin, and Markus Kessler. Complex topological features of reservoirs shape learning performances in bio-inspired recurrent neural networks. URL http://arxiv.org/abs/2211.00161. Nima Dehghani. Evolutionary Optimization Reveals Structural Constraints on Reservoir Architecture for Spatiotemporal Chaos. URL http://arxiv.org/abs/2606.22765. Joshua Faskowitz, Xiaoran Yan, Xi-Nian Zuo, and Olaf Sporns. Weighted Stochastic Block Models of the Human Connectome across the Life Span. 8(1):12997. ISSN 2045- 2322. doi: 10.1038/s41598-018-31202-1. URL https://w.nature.com/articles/ s41598-018-31202-1. Claudio Gallicchio and Alessio Micheli. Reservoir Topology in Deep Echo State Networks. volume 11731, pages 62–75. doi: 10.1007/978-3-030-30493-5 6. URL http://arxiv. org/abs/1909.11022. Daniel J. Gauthier, Erik Bollt, Aaron Griffith, and Wendson A. S. Barbosa. Next generation reservoir computing. 12(1):5564. ISSN 2041-1723. doi: 10.1038/s41467-021-25801-2. URL https://doi.org/10.1038/s41467-021-25801-2. Helmut Hauser, Auke Jan Ijspeert, Rudolf Marcel F ̈uchslin, Rolf Pfeifer, and Wolfgang Maass. Towards a theoretical foundation for morphological computation with compliant bodies. Biological Cybernetics, 105(5):355–370, 2011. doi: 10.1007/s00422-011-0471-0. Paul W. Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. Stochastic blockmod- els: First steps. Social Networks, 5(2):109–137, 1983. ISSN 0378-8733. doi: https://doi. org/10.1016/0378-8733(83)90021-7. URL https://w.sciencedirect.com/science/ article/pii/0378873383900217. Harold Hotelling. Relations between two sets of variates. Biometrika, 28(3/4):321–377, 1936. ISSN 00063444. URL http://w.jstor.org/stable/2333955. 10 WSBM-based ESN Comparison Ann Huang, Mitchell Ostrow, Satpreet H. Singh, Leo Kozachkov, Ila Fiete, and Kanaka Ra- jan. InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics. URL https://arxiv.org/abs/2510.25943v2. Herbert Jaeger.The”echo state”approach to analysing and training recur- rent neural networks.2001.URL https://w.ai.rug.nl/minds/uploads/ EchoStatesTechRepErratum.pdf. Herbert Jaeger, Mantas Lukoˇseviˇcius, Dan Popovici, and Udo Siewert. Optimization and ap- plications of echo state networks with leaky- integrator neurons. Neural Networks, 20(3): 335–352, 2007. ISSN 0893-6080. doi: https://doi.org/10.1016/j.neunet.2007.04.016. URL https://w.sciencedirect.com/science/article/pii/S089360800700041X. Echo State Networks and Liquid State Machines. Simon Kornblith, Mohammad Norouzi, Honglak Lee, and Geoffrey Hinton. Similarity of neural network representations revisited, 2019. URL https://arxiv.org/abs/1905. 00414. Nikolaus Kriegeskorte, Marieke Mur, and Peter Bandettini. Representational similarity analysis - connecting the branches of systems neuroscience. Front. Syst. Neurosci., 2:4, November 2008. Manjari S. Kulkarni and Christof Teuscher. Memristor-based reservoir computing. In Proceedings of the 2012 IEEE/ACM International Symposium on Nanoscale Architec- tures, NANOARCH ’12, page 226–232, New York, NY, USA, 2012. Association for Computing Machinery. ISBN 9781450316712. doi: 10.1145/2765491.2765531. URL https://doi.org/10.1145/2765491.2765531. Janne K. Lappalainen, Fabian D. Tschopp, Sridhama Prakhya, Mason McGill, Aljoscha Nern, Kazunori Shinomiya, Shin-ya Takemura, Eyal Gruntman, Jakob H. Macke, and Srinivas C. Turaga. Connectome-constrained networks predict neural activity across the fly visual system. Nature, 634(8036):1132–1140, Oct 2024. ISSN 1476-4687. doi: 10.1038/ s41586-024-07939-3. URL https://doi.org/10.1038/s41586-024-07939-3. Corentin L ́eger, Gautier Hamon, Eleni Nisioti, Xavier Hinaut, and Cl ́ement Moulin-Frier. Evolving reservoirs for meta reinforcement learning. In Stephen Smith, Jo ̃ao Correia, and Christian Cintrano, editors, Applications of Evolutionary Computation, pages 36–60, Cham, 2024. Springer Nature Switzerland. ISBN 978-3-031-56855-8. Mantas Lukoˇseviˇcius. A Practical Guide to Applying Echo State Networks. In Gr ́egoire Montavon, Genevi`eve B. Orr, and Klaus-Robert M ̈uller, editors, Neural Networks: Tricks of the Trade, volume 7700, pages 659–686. Springer Berlin Heidelberg. ISBN 978-3- 642-35288-1 978-3-642-35289-8. doi: 10.1007/978-3-642-35289-8 36. URL http://link. springer.com/10.1007/978-3-642-35289-8_36. Sean L. McDaniel, Marisel Villafane-Delgado, and Erik C. Johnson. Investigating Echo State Network Performance with Biologically-Inspired Hierarchical Network Structure. doi: 10.1109/IJCNN55064.2022.9892248. URL https://sci-hub.vn/10.1109/IJCNN55064. 2022.9892248. 11 Mai Clarke Filip Milisav, Andrea I. Luppi, Laura E. Su ́arez, Guillaume Lajoie, and Bratislav Misic. Neuromorphic hierarchical modular reservoirs. Nature Communications, 17(1):7962, Jun 2026. ISSN 2041-1723. doi: 10.1038/s41467-026-74466-2. URL https://doi.org/10. 1038/s41467-026-74466-2. Mitchell Ostrow, Adam Eisen, Leo Kozachkov, and Ila Fiete. Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis. URL http://arxiv.org/abs/2306.10168. Anastasiya Salova and Istv ́an A Kov ́acs. Combined topological and spatial constraints are required to capture the structure of neural connectomes. Netw. Neurosci., 9(1):181–206, March 2025. Jianghong Shi, Bryan Tripp, Eric Shea-Brown, Stefan Mihalas, and Michael A. Buice. Mousenet: A biologically constrained convolutional neural network model for the mouse visual cortex. PLOS Computational Biology, 18(9):1–30, 09 2022. doi: 10.1371/journal. pcbi.1010427. URL https://doi.org/10.1371/journal.pcbi.1010427. Kristof Vandoorne, Wouter Dierckx, Benjamin Schrauwen, David Verstraeten, Roel Baets, Peter Bienstman, and Jan Van Campenhout. Toward optical signal processing using photonic reservoir computing. Opt Express, 16(15):11182–11192, July 2008. D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt. An experimental unification of reservoir computing methods. Neural Networks, 20(3):391–403, 2007. ISSN 0893-6080. doi: https://doi.org/10.1016/j.neunet.2007.04.003. URL https://w.sciencedirect. com/science/article/pii/S089360800700038X. Echo State Networks and Liquid State Machines. David Verstraeten, Benjamin Schrauwen, and Dirk Stroobandt.Reservoir computing with stochastic bitstream neurons. 2005. URL https://api.semanticscholar.org/ CorpusID:7506801. Appendix A. Preliminaries A.1. Echo State Network Echo State Networks (ESNs) (Jaeger, 2001) provide a recurrent neural network architecture where temporal dynamics are modeled using a fixed, nonlinear hidden layer, termed the reservoir. By restricting training to the output readout layer via linear regression, ESNs avoid the computational expense of backpropagation through time. Let s(t) ∈R N u denote the input vector and V (t) ∈R N x denote the reservoir state at discrete time t, where N x is the number of reservoir units. In a standard ESN, the reservoir topology is defined by a sparse random graph, often instantiated as an Erd ̋os–R ́enyi network with connection probability p ER . The state evolution of the reservoir is governed by the dynamical equation: V (t + δt) = (1− α)V (t) + αf (γW in s(t) + ρWV (t)), 12 WSBM-based ESN Comparison where α = δt/τ ∈ (0, 1] represents the leakage rate governed by the integration timescale τ . The function f (·) is an element-wise nonlinear activation (e.g tanh function), W in ∈R N x ×N u is the input weight matrix, and W ∈R N x ×N x is the fixed recurrent weight matrix defining the network’s internal connectivity. The scalar multipliers γ and ρ dictate the input scaling and recurrent scaling, respectively. For stable functioning, the dynamical system must satisfy the echo state property, which asserts that the asymptotic state of the reservoir depends on the history of the driving input s(t), independent of initial conditions: lim t→∞ ∥V (t,s,V 0 )− V (t,s,V ′ 0 )∥ = 0. Hence, it is necessary that the recurrent weight matrix W got its spectral radius scaled, max i |λ i (ρW )| < 1, with λ i (ρW ) denote the i-th eigenvalue of the scaled recurrent weight matrix in the complex plane. (Jaeger et al., 2007; Lukoˇseviˇcius). A.2. Stochastic Block Model The Stochastic Block Model (SBM) (Holland et al., 1983; Bridgeford et al., 2025) is a generative probabilistic model formulated to infer latent community structures within a network. Let a network of N nodes be represented by an adjacency matrix A ∈R N×N . The SBM partitions the node set into K disjoint blocks, with the latent block assignment for node i denoted by z i ∈ 1,...,K. The premise of the SBM is stochastic equivalence: the probability of an edge between any two nodes depends exclusively on their respective block assignments. Formally, the existence of a binary edge A ij ∈ 0, 1 is modeled as a conditionally independent Bernoulli random variable: A ij | z i ,z j ∼ Bernoulli(θ z i ,z j ), where Θ ∈ [0, 1] K×K is the inter-block connection probability matrix, and θ z i ,z j defines the probability of an edge forming between a node in block z i and a node in block z j . A limitation of the standard SBM is its restriction to binary topologies, which excludes the parameterization of interaction strengths inherent to empirical connectomes. The Weighted Stochastic Block Model (WSBM) (Aicher et al., b,a) resolves this by extending the frame- work to jointly model the topological structure and continuous edge weights. Under the WSBM formulation, the conditional distribution of an edge weight is parameterized via a univariate exponential family: A ij | z i ,z j ∼ ExponentialFamily(η z i ,z j ), whereη ∈R K×K represents the matrix of natural parameters, such that its element η z i ,z j governs the weight distribution between communities z i and z j . This is expressed via the probability density function P (A ij | z i ,z j ) = h(A ij ) exp(η T z i ,z j T (A ij ) − c(η z i ,z j )), where T (·) is the sufficient statistic and h(·) is the base measure. This extension provides the formal apparatus required to map the continuous, meso-scale organization of biological connectomes into the topological priors W of the aforementioned Echo State Networks. 13 Mai Clarke Appendix B. Hyperparameter space and failure mode Figure 5: Hyperparameters space selection. Figure 6: Failure mode of InputDSA on each instance of 5 classes on Mackey-Glass time- series due to input’s low-dimensionality vs OU noise timeseries. 14 WSBM-based ESN Comparison Figure 7: The performance of agents across Ant-v4 and HalfCheetah-v4 tasks with its cor- responding top and bottom 10 dynamics Figure 8: Failure mode of using N4SID on Swimmer InputDSA. 15 Mai Clarke Appendix C. C.1. Algorithm Input: Number of nodes N , communities K, target spectral radius ρ, negative edge prob- ability p neg , community assignment logic. Block matrices for means Ω ∈R K×K , standard deviations Σ∈R K×K + , and connection probabilities Θ∈ [0, 1] K×K . Output: Initialized reservoir weight matrix W res ∈R N×N , community assignment vector z ∈1,...,K N . 1. Determine community assignments (a) z ← Equal distribution, Dirichlet distribution, or explicit array configuration mapping each node i to a community k. 2. Sample initial states and connectivity masks (a) W ij ∼N (Ω z i ,z j , Σ 2 z i ,z j ) for all i,j ∈1,...,N (b) M ij ∼ Bernoulli(Θ z i ,z j ) for all i,j ∈1,...,N 3. Enforce undirected topology and self-connection constraints (a) W ij ← W ij · M ij for i < j (b) W ji ← W ij for i < j (c) W i ← 0 for all i∈1,...,N 4. Apply structural perturbations (a) F ij ∼ Bernoulli(p neg ) for i < j (b) W ij ←−|W ij | and W ji ←−|W ij | for all i,j | i < j∧ F ij = 1 5. Scale to target spectral dynamics (a) λ max ← max|eigvals(W )| (b) W res ← ρ W λ max C.2. RL Loop 16 WSBM-based ESN Comparison RL algorithm Reservoir Update Actor-Critic MLP Environment Figure 9: A diagram features the flow of information during a RL task, adapted from L ́eger et al. (2024) 17 Mai Clarke 18