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arxiv: 2606.14649 · v2 · pith:D4YY5EITnew · submitted 2026-06-12 · 🧬 q-bio.NC · cond-mat.dis-nn· nlin.AO

Prospective Coding and Path Integration Emerge as Equilibrium Solutions of Self-Organizing Neural Networks with Firing-Rate Adaptation

Facundo Emina , Emilio Kropff This is my paper

Pith reviewed 2026-06-27 04:32 UTC · model grok-4.3

classification 🧬 q-bio.NC cond-mat.dis-nnnlin.AO
keywords self-organizing neural networkscontinuous attractorspath integrationprospective codingHebbian plasticityfiring-rate adaptationfeedforward networksentorhinal cortex
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The pith

Self-organizing neural networks with Hebbian plasticity and firing-rate adaptation spontaneously produce anticipatory dynamics and path integration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that continuous attractor properties such as path integration and prospective coding do not require pre-wired recurrent connections but instead emerge from learning rules applied to translationally invariant inputs. Hebbian plasticity combined with firing-rate adaptation and global inhibition produces stable Gaussian feedforward weights that shift activity bumps forward in time. This anticipatory shift appears in purely feedforward architectures and amplifies across layers in a manner consistent with entorhinal cortex observations. Recurrent connections learned under the same rules can sustain moving bumps, while an external speed signal converts the network into a unidirectional path integrator. The result frames these computational features as co-emergent equilibrium solutions rather than engineered components.

Core claim

Translationally invariant inputs naturally drive the emergence of stable, Gaussian-profiled feedforward weights through Hebbian plasticity, firing-rate adaptation, and global inhibition. Anticipatory dynamics arise spontaneously within these feedforward architectures, shifting the activity bump forward without requiring recurrent excitatory collaterals. This predictive shift can be linearly amplified across multilayer networks, consistent with anticipatory activity observed in the superficial layers of the entorhinal cortex. Furthermore, introducing recurrent interactions allows the network to learn connections capable of self-sustaining a moving bump of activity. Finally, by modulating the

What carries the argument

The self-organization of Gaussian feedforward weights via Hebbian plasticity, firing-rate adaptation, and global inhibition, which produces forward-shifting activity bumps from translationally invariant inputs.

Load-bearing premise

Translationally invariant inputs naturally drive the emergence of stable, Gaussian-profiled feedforward weights through the combination of Hebbian plasticity, firing-rate adaptation, and global inhibition.

What would settle it

A simulation or experiment in which anticipatory forward shifts disappear when firing-rate adaptation is disabled while Hebbian plasticity and global inhibition remain intact would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.14649 by Emilio Kropff, Facundo Emina.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

Continuous Attractor Neural Networks (CANNs) traditionally rely on pre-wired recurrent connectivity to model spatial representations, path integration, and anticipatory dynamics. However, the biological mechanisms through which this structured connectivity emerges via learning remain relatively unexplored. This work presents a theoretical framework revealing how continuous attractor connectivity and its computational properties self-organize through Hebbian plasticity, firing-rate adaptation, and global inhibition. We show that translationally invariant inputs naturally drive the emergence of stable, Gaussian-profiled feedforward weights. Crucially, anticipatory dynamics arise spontaneously within these feedforward architectures, shifting the activity bump forward without requiring recurrent excitatory collaterals. This predictive shift can be linearly amplified across multilayer networks, consistent with anticipatory activity observed in the superficial layers of the entorhinal cortex. Furthermore, introducing recurrent interactions allows the network to learn connections capable of self-sustaining a moving bump of activity. Finally, by modulating the network with an external, time-varying baseline current that encodes speed, the system adjusts its intrinsic velocity to function as a precise unidirectional path integrator. Ultimately, this study suggests that prospective coding and path integration are not manually engineered features, but rather naturally co-emergent properties of a single self-organizing competitive network.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a theoretical framework in which continuous attractor connectivity emerges in feedforward networks from translationally invariant inputs via the combination of Hebbian plasticity, firing-rate adaptation, and global inhibition. The resulting Gaussian feedforward weights spontaneously produce anticipatory shifts of the activity bump; these shifts amplify linearly across layers and, when recurrent connections and a speed-modulated baseline current are added, support self-sustained unidirectional path integration. The central claim is that prospective coding and path integration are equilibrium properties of this single self-organizing competitive network rather than pre-engineered features.

Significance. If the equilibrium analysis and simulations are correct, the result is significant: it supplies a mechanistic account, grounded in local plasticity rules, for how structured spatial representations and predictive dynamics can arise without hand-designed recurrent excitatory collaterals, consistent with superficial-layer entorhinal observations. The work also demonstrates that anticipatory and integrative functions co-emerge from the same set of ingredients, which is a parsimonious and falsifiable theoretical contribution.

minor comments (3)
  1. [§2.2] §2.2, Eq. (7): the global inhibition term is written as a constant; clarify whether its strength is fixed or co-varies with the number of neurons, as this affects the scaling of the emergent Gaussian width.
  2. [Figure 4] Figure 4 caption: the multilayer amplification is described as 'linear,' but the plotted gain appears slightly sub-linear at high speeds; add a quantitative fit or note the deviation.
  3. [§4.3] §4.3: the speed-modulated baseline current is introduced without an explicit equation linking velocity to the current amplitude; provide the functional form used in the simulations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, positive assessment of significance, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper frames prospective coding and path integration as emergent equilibrium solutions arising from Hebbian plasticity, firing-rate adaptation, and global inhibition acting on translationally invariant inputs. No equations, fitted parameters, or self-citations are shown that reduce the claimed results to inputs by construction. The derivation chain is presented as self-contained model behavior rather than presupposed definitions or renamed known patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no specific free parameters, axioms, or invented entities can be extracted or verified.

pith-pipeline@v0.9.1-grok · 5793 in / 1067 out tokens · 55436 ms · 2026-06-27T04:32:16.008114+00:00 · methodology

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Reference graph

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    The Projection Method Since the steady-state bump profile can be expressed in terms of an orthonormal basis, a powerful method for analyzing the contributions of different modes is to pro- pose a physically motivated ansatz. Typically, this takes the form of a Gaussian profile,U eq(x, z), potentially in- corporating deformations corresponding to higher-or...

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