{"id":3219,"date":"2026-01-08T11:44:38","date_gmt":"2026-01-08T11:44:38","guid":{"rendered":"https:\/\/beyondtheimpact.net\/?p=3219"},"modified":"2026-01-08T11:44:38","modified_gmt":"2026-01-08T11:44:38","slug":"learning-rules-for-signals-that-arrive-backward","status":"publish","type":"post","link":"https:\/\/beyondtheimpact.net\/?p=3219","title":{"rendered":"Learning rules for signals that arrive backward"},"content":{"rendered":"

<\/a><\/p>\n

Temporal credit assignment becomes especially nontrivial when signals are permitted to propagate backward through a network, because the apparent order of cause and effect is partially inverted. In standard forward-processing systems, activity flows from inputs to outputs, and learning rules can rely on local correlations between presynaptic and postsynaptic activity to infer which synapses contributed to a specific outcome. When signal flow is reversed, however, postsynaptic activity or output errors can arrive at earlier processing stages after the original forward signal has decayed, raising the question of how the system determines which past events should be modified. This challenge resembles a temporal bookkeeping problem: the system must reconstruct which internal states, occurring at earlier times, were causally responsible for a downstream event that is now sending a backward influence.<\/p>\n

One way to conceptualize temporal credit assignment in reversed flow is through a comparison with conventional backpropagation. In backpropagation, error signals are mathematically propagated from outputs to inputs along the same connections used in the forward pass, but in a purely algorithmic, not physical, sense. The network maintains an ordered record of intermediate activations so that each weight can be updated according to its contribution to the final error. In contrast, when signals physically or functionally travel backward in a biological or neuromorphic system, intermediate states may not be explicitly stored. Instead, neurons and synapses must rely on implicit traces, such as residual activity, short-term synaptic dynamics, or molecular markers, to bridge the temporal gap between the forward event and the later-arriving backward signal. Temporal credit assignment then becomes a problem of aligning these internal traces with backward-propagating influences to define a usable learning signal.<\/p>\n

Reversed signal flow suggests a form of temporal retrocausality in which later network states appear to influence earlier synaptic modifications. From the perspective of an external observer, a postsynaptic neuron could emit a backward-going signal once its firing pattern is evaluated against a behavioral outcome or prediction error. This backward signal interacts with stored traces of the neuron’s prior inputs, effectively assigning credit (or blame) to synapses that were active at some earlier time. Although physical causality is never actually violated\u2014the backward signal is triggered only after the outcome is known\u2014the computational viewpoint treats the learning signal as if it were informing past synaptic decisions. The crucial requirement is that the system maintain a time-locked representation of past activity that remains accessible when the backward signal arrives, otherwise no consistent mapping between outcomes and contributing causes can be established.<\/p>\n

An important design choice concerns how long these activity traces persist and how they decay over time. If traces decay too rapidly, backward signals will find no reliable record of which synapses were involved, leading to noisy or misassigned updates. If traces are too persistent, the system risks smearing responsibility over many irrelevant events, diluting the specificity of credit assignment. This implies that temporal credit assignment in reversed signal flow requires a balance between stability and flexibility of memory traces. Mechanisms such as eligibility traces, in which synapses transiently mark themselves as \u201celigible\u201d for modification after particular patterns of pre- and postsynaptic activity, offer a way to store time-localized information until a delayed teaching or error signal arrives. When the backward signal finally reaches a neuron or synapse, these eligibility traces can be converted into actual plasticity, selectively reinforcing or weakening the synapses that were causally associated with the eventual outcome.<\/p>\n

In this framework, the direction of signal propagation and the direction of causal responsibility are partially decoupled. The forward pass carries sensory evidence and internal states toward an output, while the backward pass carries evaluative information about how good or bad the output was relative to some target or expectation. Temporal credit assignment in reversed flow therefore hinges on a matching between two temporal sequences: the history of local activity, summed up in eligibility or other forms of hidden state, and the later arrival of a modulatory or error-related signal. Learning rules are then defined over pairs consisting of a backward signal at time t and an eligibility trace representing activity at a prior time t \u2212 \u0394. Through this pairing, the system can update parameters as though it were retroactively correcting the computations that led to the outcome, even though all operations strictly obey forward-in-time causality.<\/p>\n

Such an arrangement naturally connects with the notion of the brain as a bayesian brain performing probabilistic inference over time. In predictive processing frameworks, neural circuits continuously compare incoming sensory data against internal predictions shaped by priors. Prediction error signals are thought to propagate through the hierarchy, potentially in both feedforward and feedback directions, to update beliefs about hidden causes. When these error signals travel backward to earlier levels of processing, temporal credit assignment must specify which synapses encoding priors or generative models should be adjusted. Backward-propagating errors encapsulate information about discrepancies between predicted and observed inputs at a later time, yet they must modulate synapses that influenced predictions at an earlier time. Therefore, the system must maintain a temporally structured representation of generative activity that can be selectively modified in light of new evidence communicated by backward signals.<\/p>\n

Temporal structure also raises the question of how multiple outcomes, potentially separated by varying delays, can be associated with overlapping patterns of past activity. If a particular subset of synapses participates in multiple computations across time, then backward signals corresponding to different outcomes may arrive while earlier eligibility traces are still partially active. This can lead to interference, where updates driven by one outcome contaminate those needed for another. Effective temporal credit assignment in reversed flow requires strategies for disambiguating overlapping traces, such as context-dependent gating, separate time constants for different classes of synapses, or compartmentalized processing that segregates signals associated with distinct tasks or episodes. These mechanisms allow the network to preserve the specificity of learning despite the temporal entanglement of signals and outcomes.<\/p>\n

Another critical aspect concerns the relative timing between the backward signal and the forward-derived eligibility trace. In many plausible mechanisms, plasticity depends on a precise temporal relationship, not merely on the presence of both signals. If the backward signal arrives too early, before the forward activity has been encoded into a stable trace, the relevant synapses may not yet be marked as eligible for modification. If it arrives too late, the trace may have decayed beyond a usable threshold. This sensitivity to timing can endow the system with a form of temporal selectivity, where synapses that consistently participate in computations leading to reliably rewarded outcomes accumulate stronger, more durable traces. Over repeated experiences, the temporal matching between cause and evaluative feedback sharpens, yielding an internal alignment that supports stable, task-relevant learning despite the reversed propagation of teaching signals.<\/p>\n

From a systems perspective, temporal credit assignment in reversed signal flow implies that neural circuits must integrate three partially independent streams of information: the forward sensory or internal drive, the backward evaluative or error signals, and the internal dynamics that store or transform activity into an update-ready format. The interaction among these streams determines which synapses undergo plasticity at any given moment. By tuning the temporal properties of eligibility traces, the sensitivity of neurons to backward signals, and the global modulatory context in which these processes occur, a network can carve out a robust mapping from temporally distant outcomes to the synapses that enabled them. This mapping is what allows backward-propagating influences to shape learning without explicit global supervision or a complete replay of the original inputs, effectively implementing a biologically plausible counterpart to temporal backpropagation.<\/p>\n

Biologically plausible mechanisms for backward learning<\/h3>\n

Biological neural tissue offers several candidate mechanisms through which information about outcomes can flow backward and interact with stored traces of past activity. One of the most direct substrates for backward learning is the ubiquitous system of feedback and recurrent connections in the brain. Cortical and subcortical circuits rarely operate as purely feedforward pipelines; instead, they form dense loops in which higher-order areas project back to earlier ones. These feedback pathways can carry evaluative or contextual information derived from later processing stages, including signals related to reward, surprise, or mismatch between expectation and observation. When such feedback arrives, it can modulate the excitability of upstream neurons, alter their ongoing dynamics, and, critically, gate synaptic plasticity at the synapses that had been active slightly earlier in time.<\/p>\n

Within this anatomical scaffold, dendritic integration provides a powerful mechanism for implementing directionally specific and temporally sensitive backward learning. Pyramidal neurons, for example, receive feedforward input primarily onto their basal dendrites, while feedback and modulatory inputs often target the apical dendrites or tuft regions. This spatial segregation allows a kind of internal comparison between \u201cbottom-up\u201d evidence and \u201ctop-down\u201d evaluative or predictive signals. When a backward-propagating signal arrives on apical dendrites, it can trigger local calcium spikes, backpropagating action potentials, or other nonlinear events that selectively amplify or suppress plasticity at recently active basal synapses. In this way, feedback does not need to replay the original sensory pattern; instead, it modulates the transformation of preexisting eligibility traces into lasting synaptic changes.<\/p>\n

Eligibility traces themselves can be implemented through a variety of short-lived biochemical and biophysical events. Transient phosphorylation states, calcium concentrations, activation of second-messenger cascades, and presynaptic vesicle dynamics can all store graded information about recent pre- and postsynaptic activity. When a synapse experiences a particular temporal pattern of spikes, it may enter a metastable state that does not immediately alter synaptic strength but instead marks it as \u201cprimed\u201d for future modification. If, within a critical time window, a backward signal associated with reward, punishment, or error arrives at the neuron, neuromodulators and dendritic signals can convert the primed state into a concrete change in synaptic efficacy. This separation between eligibility creation and eligibility resolution allows learning rules to exploit backward signals without requiring them to be perfectly synchronized with the original activity.<\/p>\n

Neuromodulatory systems play a central coordinating role in this process, providing a broadcast channel for global or semi-global backward signals that convey information about behavioral outcomes. Dopamine, for instance, is widely studied as a carrier of reward prediction error signals, reflecting the difference between observed and expected outcomes. When dopaminergic neurons fire in response to an unexpected reward or the omission of an expected reward, their axonal projections deliver a phasic signal to large portions of the brain. Synapses that were recently active and have formed eligibility traces can now be strengthened or weakened depending on the sign and magnitude of this modulatory pulse. Similar roles are played by other neuromodulators such as norepinephrine, serotonin, and acetylcholine, which can encode surprise, uncertainty, effort, or attentional significance. Together, these systems provide biologically plausible pathways for backward-propagating evaluative information to interact with temporally displaced activity patterns.<\/p>\n

Spike-timing-dependent plasticity (STDP) offers another plausible substrate for implementing backward learning in time. In classical STDP, the direction and magnitude of synaptic change depend on the relative timing of pre- and postsynaptic spikes: presynaptic spikes that precede postsynaptic spikes within a characteristic window tend to lead to potentiation, whereas the reversed order tends to induce depression. Backward-propagating action potentials from the soma into dendrites serve as the postsynaptic timing signal, effectively conveying information about the neuron\u2019s later firing back to synapses that were activated slightly earlier. By extending or modulating STDP rules with neuromodulatory signals and dendritic compartmentalization, it becomes possible to implement more complex forms of temporal credit assignment where postsynaptic responses and behavioral outcomes occurring after an input can selectively reshape the synapses that generated those responses.<\/p>\n

Compartment-specific dynamics allow neurons to implement a local form of error separation that bears a loose analogy to backpropagation while remaining grounded in known physiology. Basal dendrites, receiving feedforward input, can be treated as encoding the \u201cprediction\u201d or internal representation based on earlier layers, while apical compartments can encode feedback signals that convey something like an error or teaching signal. Nonlinear interactions between these compartments, such as calcium spikes triggered only when apical and basal inputs coincide within a certain window, can define conditions under which plasticity is allowed to proceed. This creates a gating mechanism such that synaptic modifications are preferentially applied when there is both evidence of prior engagement in a computation (via basal activation and eligibility traces) and evidence from later stages that the outcome of that computation was significant or mismatched relative to internal priors.<\/p>\n

Astrocytes and other glial cells add an additional layer of temporal integration and spatial coordination that can support backward learning. Astrocytes envelop synapses and can monitor activity patterns across local microcircuits through calcium waves and neurotransmitter uptake. They also regulate the local extracellular milieu, including the concentration of neuromodulators and metabolic resources. By responding to bursts of activity associated with outcomes or global states\u2014such as stress, attention, or arousal\u2014astrocytes can modulate when and where plasticity is expressed. This modulation can retroactively amplify changes at synapses that were active during a recently completed behavioral episode, effectively implementing a coarse form of backward credit assignment across extended spatial and temporal scales.<\/p>\n

Recurrent microcircuits, such as those found in cortical columns, basal ganglia loops, and cerebellar circuits, provide structured substrates in which backward signals can circulate and repeatedly interact with older traces. In cortico-basal ganglia-thalamo-cortical loops, for example, reward-related signals influence the selection and reinforcement of action patterns, while recurrent projections enable information about selected actions and their outcomes to reverberate through upstream cortical areas. Within these loops, backward influences need not follow a single anatomical path; instead, they are distributed across multiple convergent and divergent projections. This redundancy allows outcome-related signals to find and modulate many of the synaptic pathways that previously contributed to a decision, even when the original activity has ceased.<\/p>\n

In sensory hierarchies, feedback connections from higher to lower areas frequently carry expectation and prediction signals shaped by experience. When these predictions fail\u2014because incoming data conflict with internal models\u2014prediction error signals can be generated and propagate both forward and backward. A biologically plausible backward learning mechanism in this context is that synapses contributing to incorrect predictions become tagged by local inhibitory or modulatory signals. Subsequent feedback carrying error information can then target these tagged synapses for weakening, while those involved in accurate predictions can be stabilized or strengthened. Over time, this interaction between prediction, error, and selective plasticity allows the network to refine its generative models using backward-directed information flow without requiring an explicit, algorithmic form of backpropagation.<\/p>\n

Timing relationships between neuromodulatory bursts and local circuit activity are crucial for these mechanisms to operate effectively. Many neuromodulatory systems have firing patterns that are tightly coupled to key behavioral events such as outcome delivery, novelty detection, or shifts in task demands. Because eligibility traces at synapses decay over time, the lag between an event and its evaluative signal defines which synapses can be updated. For instance, if a reward occurs several hundred milliseconds after a decision, dopaminergic bursts arriving during that window will preferentially interact with traces formed during the decision and pre-decision period. Longer delays may require multi-stage mechanisms, such as working-memory circuits or hippocampal replay, to maintain or reconstruct the patterns of activity that will later be subject to outcome-based modification.<\/p>\n

Hippocampal and cortical replay during sleep and quiet wakefulness offers a complementary route for backward learning when immediate neuromodulatory feedback is not available. In replay, sequences of neural activity that occurred during behavior are partially reinstated, often at compressed timescales. Outcome-related signals, including those arising from consolidation processes and global neuromodulatory tone, can act on these replayed sequences to refine synaptic weights as though the outcome information were traveling backward in time to meet the original patterns. This mechanism sidesteps strict temporal constraints by decoupling learning from real-time behavior while still allowing backward-like influences to shape earlier computations embedded in memory traces.<\/p>\n

Local inhibitory circuits contribute to backward learning by sculpting the window in which plasticity is permissible. Interneurons can dynamically gate the flow of both feedforward and feedback signals, controlling whether a neuron is in a plastic or nonplastic state. For example, feedback arriving from higher areas might preferentially recruit specific classes of inhibitory interneurons that suppress or permit dendritic spikes in particular compartments. By linking the activation of these interneurons to outcome-related or context-specific signals, the brain can restrict synaptic changes to episodes where backward-propagating information indicates that an adjustment is warranted. This gating reduces unwanted interference between overlapping episodes and helps ensure that plasticity is deployed selectively along the pathways most relevant to future performance.<\/p>\n

Altogether, these mechanisms suggest a picture in which backward learning in biological systems relies on an interplay of eligibility traces, feedback projections, dendritic segregation, neuromodulation, replay, and inhibitory control. Rather than computing precise gradient signals as in conventional backpropagation, the brain appears to approximate temporal credit assignment through layered, heterogeneous processes that align delayed evaluative signals with stored representations of past activity. Such arrangements preserve physical causality while enabling a functional form of retrocausality at the algorithmic level, where later outcomes effectively reach back to sculpt the synapses that contributed to earlier decisions and predictions.<\/p>\n

Mathematical formulation of retrocausal update rules<\/h3>\n

To render backward-propagating signals into concrete changes of synaptic strength, it is useful to formalize retrocausal learning rules in a common mathematical framework. Consider a network with neurons indexed by i, j, and synaptic weights wij<\/sub> from neuron j to neuron i. Forward activity at time t is denoted by xj<\/sub>(t), and postsynaptic activity by yi<\/sub>(t). A backward or evaluative signal arriving at neuron i at time t is written as bi<\/sub>(t); this can represent an error, a reward prediction error, or a more general feedback signal. The key ingredient that bridges forward and backward events is an eligibility trace eij<\/sub>(t), a temporally extended variable that encodes the recent history of pre- and postsynaptic interactions at synapse (i, j). Learning rules in this setting typically take the form<\/p>\n

\u0394wij<\/sub>(t) = \u03b7 \u00b7 bi<\/sub>(t) \u00b7 eij<\/sub>(t),<\/p>\n

where \u03b7 is a learning rate. This structure separates the creation of eligibility (driven by forward activity) from its resolution (driven by backward, outcome-related signals). The apparent retrocausality arises because eij<\/sub>(t) was shaped by events at times t\u2032 < t, yet the actual plasticity update is applied later, at time t, when bi<\/sub>(t) becomes available.<\/p>\n

Eligibility traces can be defined in many ways, but a common choice is an exponentially decaying memory of a local plasticity-driving signal Fij<\/sub>(t):<\/p>\n

\u03c4e<\/sub> (deij<\/sub>\/dt) = \u2212eij<\/sub>(t) + Fij<\/sub>(t),<\/p>\n

with time constant \u03c4e<\/sub>. The function Fij<\/sub>(t) encodes how pre- and postsynaptic activity would influence synaptic change if a teaching signal were present immediately. For example, a basic Hebbian form is Fij<\/sub>(t) = xj<\/sub>(t) yi<\/sub>(t), while a spike-timing-dependent plasticity-inspired form might include terms that depend on the precise temporal order of spikes. The exponential kernel ensures that past activity affects the present eligibility with a weight that decays approximately as exp(\u2212\u0394t\/\u03c4e<\/sub>), where \u0394t is the delay between the original event and the current time. In discrete time, with step size \u0394t, an equivalent update is<\/p>\n

eij<\/sub>(t + \u0394t) = (1 \u2212 \u0394t\/\u03c4e<\/sub>) eij<\/sub>(t) + Fij<\/sub>(t).<\/p>\n

Combining these equations, the weight dynamics can be written as a two-stage process: first, eligibility accumulation driven by local activity, and second, eligibility conversion driven by backward signals:<\/p>\n

eij<\/sub>(t + \u0394t) = (1 \u2212 \u0394t\/\u03c4e<\/sub>) eij<\/sub>(t) + Fij<\/sub>(t),
\nwij<\/sub>(t + \u0394t) = wij<\/sub>(t) + \u03b7 \u00b7 bi<\/sub>(t) \u00b7 eij<\/sub>(t).<\/p>\n

This factorization clarifies why retrocausal update rules do not violate ordinary causality. The forward activity at earlier times only influences the future through the causal evolution of eij<\/sub>(t). When bi<\/sub>(t) finally arrives, it selectively amplifies or suppresses the portion of eij<\/sub>(t) arising from specific time intervals, effectively reweighting the influence of past events on present plasticity.<\/p>\n

To relate these retrocausal rules to more familiar gradient-based methods such as backpropagation, consider a loss function L that depends on network outputs at discrete times tk<\/sub>. In standard backpropagation through time, weight updates are computed as<\/p>\n

\u0394wij<\/sub> = \u2212\u03b7 \u2202L\/\u2202wij<\/sub> = \u2212\u03b7 \u03a3t<\/sub> (\u2202L\/\u2202yi<\/sub>(t)) (\u2202yi<\/sub>(t)\/\u2202wij<\/sub>).<\/p>\n

The term \u2202L\/\u2202yi<\/sub>(t) plays the role of a backward error signal, while \u2202yi<\/sub>(t)\/\u2202wij<\/sub> encapsulates how the synapse affects the neuron\u2019s activity, usually involving products of pre- and postsynaptic quantities propagated through time. Retrocausal learning rules approximate this gradient decomposition by identifying bi<\/sub>(t) \u2248 \u2212\u2202L\/\u2202yi<\/sub>(t) and eij<\/sub>(t) \u2248 an online approximation of \u2202yi<\/sub>(t)\/\u2202wij<\/sub>. The major difference is that, instead of requiring full knowledge of the global computational graph, eij<\/sub>(t) is computed locally from observable variables such as spikes, membrane potentials, and short-term synaptic dynamics.<\/p>\n

For instance, in rate-based models where yi<\/sub>(t) = \u03c6(\u03a3j<\/sub> wij<\/sub> xj<\/sub>(t)), with activation function \u03c6, one can choose<\/p>\n

Fij<\/sub>(t) = \u03c6\u2032(hi<\/sub>(t)) xj<\/sub>(t),<\/p>\n

where hi<\/sub>(t) = \u03a3j<\/sub> wij<\/sub> xj<\/sub>(t) is the total input. Integrated over time, eij<\/sub>(t) then becomes a filtered estimate of how the synapse contributes to changes in yi<\/sub>, in line with gradient-based sensitivity. If the backward signal bi<\/sub>(t) corresponds to a prediction error, such as the difference between target and actual output at time t, then \u0394wij<\/sub>(t) \u2248 \u2212\u03b7 \u2202L\/\u2202wij<\/sub>(t) in expectation, assuming appropriate noise and stationarity conditions. This construction bridges the gap between normative gradient descent and implementable biological learning rules.<\/p>\n

In spiking neuron models, the derivation follows a similar logic but uses spike trains and membrane dynamics. Let si<\/sub>(t) denote the spike train of neuron i, and Vi<\/sub>(t) its membrane potential. A general eligibility component for synapse (i, j) can be written as<\/p>\n

Fij<\/sub>(t) = fpre<\/sub>(sj<\/sub>, t) \u00b7 fpost<\/sub>(si<\/sub>, Vi<\/sub>, t),<\/p>\n

where fpre<\/sub> and fpost<\/sub> extract relevant pre- and postsynaptic features, such as filtered spike counts or voltage-dependent terms. One example consistent with STDP-like plasticity is<\/p>\n

Fij<\/sub>(t) = A+<\/sub> \u03a3tj<\/sub>\u2208spikesj<\/sub><\/sub> \u03ba+<\/sub>(t \u2212 tj<\/sub>) si<\/sub>(t) \u2212 A\u2212<\/sub> \u03a3ti<\/sub>\u2208spikesi<\/sub><\/sub> \u03ba\u2212<\/sub>(t \u2212 ti<\/sub>) sj<\/sub>(t),<\/p>\n

with kernels \u03ba+<\/sub>, \u03ba\u2212<\/sub> encoding potentiation and depression windows. The eligibility trace then aggregates these contributions, while the backward signal bi<\/sub>(t) gates whether they should be turned into actual changes. This decoupling allows spike-timing-sensitive plasticity to be expressed only when outcome-related information is present, aligning detailed temporal patterns of activity with delayed evaluative signals.<\/p>\n

When the network is embedded in a probabilistic framework such as a bayesian brain model, the backward signals naturally take the form of prediction errors or gradients of a log-likelihood or variational free energy. Suppose hidden states z and observations o are linked by a generative model p(o, z | \u03b8), with parameters \u03b8 represented by synapses. A common objective is to maximize the log evidence log p(o | \u03b8) or minimize a related free energy functional. Stochastic gradient ascent on log p(o | \u03b8) yields updates<\/p>\n

\u0394\u03b8 \u221d \u2202\/\u2202\u03b8 log p(o | \u03b8) = Ep(z|o,\u03b8)<\/sub>[\u2202\/\u2202\u03b8 log p(o, z | \u03b8)].<\/p>\n

In neural implementations, backward-propagating signals encode discrepancies between predicted and observed activity, approximating \u2202\/\u2202\u03b8 log p(o, z | \u03b8). Retrocausal update rules implement this gradient in an online way by using eligibility traces to capture how synaptic parameters influence the generative predictions at earlier times, while feedback conveys the current mismatch between observations and those predictions. Hence bi<\/sub>(t) can be interpreted as a local proxy for a prediction error derived from priors and incoming data, and eij<\/sub>(t) as a history-dependent sensitivity measure. Their product approximates a stochastic gradient step on a probabilistic objective without requiring explicit computation of full posterior distributions.<\/p>\n

Another useful formulation arises in the context of reinforcement learning, where outcomes are expressed as scalar rewards R(t). Let \u03b4(t) denote a reward prediction error, such as in temporal-difference learning: \u03b4(t) = R(t) + \u03b3V(t + 1) \u2212 V(t), where V is a value estimate and \u03b3 a discount factor. A retrocausal synaptic update consistent with policy gradient principles can be written as<\/p>\n

\u0394wij<\/sub>(t) = \u03b7 \u03b4(t) eij<\/sub>(t),<\/p>\n

with<\/p>\n

eij<\/sub>(t + 1) = \u03bb eij<\/sub>(t) + \u2202 log \u03c0(a(t) | state(t); w)\/\u2202wij<\/sub>,<\/p>\n

where \u03c0 is a policy and \u03bb is an eligibility decay factor. Here, eij<\/sub>(t) approximates the contribution of synapse (i, j) to the log-probability of selected actions, while \u03b4(t) carries backward information about whether those actions were better or worse than expected. The mathematical structure is directly analogous to biologically plausible dopamine-gated plasticity: \u03b4(t) corresponds to dopaminergic bursts, and eij<\/sub>(t) to synapse-specific traces created during decision making. Outcome signals arriving after behavior have thus a principled way to reach back and modify synapses according to their inferred causal role.<\/p>\n

Temporal structure can also be incorporated more explicitly by indexing eligibility traces with both space and time. Let eij<\/sub>(t, \u03c4) represent the eligibility at synapse (i, j) due to activity that occurred at relative time offset \u03c4 in the past. The evolution might follow<\/p>\n

\u2202eij<\/sub>(t, \u03c4)\/\u2202t + \u2202eij<\/sub>(t, \u03c4)\/\u2202\u03c4 = \u2212eij<\/sub>(t, \u03c4)\/\u03c4e<\/sub> + Gij<\/sub>(t) \u03b4(\u03c4),<\/p>\n

where Gij<\/sub>(t) is an instantaneous plasticity drive and \u03b4 is the Dirac delta. Backward signals bi<\/sub>(t) can then weight this distributed trace according to a temporal kernel K(\u03c4),<\/p>\n

\u0394wij<\/sub>(t) = \u03b7 \u222b K(\u03c4) bi<\/sub>(t) eij<\/sub>(t, \u03c4) d\u03c4.<\/p>\n

This formulation makes explicit that plasticity depends not only on whether a synapse was active but also on when it was active relative to the arrival of backward information. Appropriate choices of K can implement preference for recent events, multi-step delays, or even non-monotonic temporal sensitivity, allowing networks to capture complex cause\u2013effect relationships that span extended intervals.<\/p>\n

Many practical retrocausal learning schemes must contend with noisy and approximate backward signals. Suppose that bi<\/sub>(t) = gi<\/sub>(t) + \u03bei<\/sub>(t), where gi<\/sub>(t) is an unbiased estimator of the true gradient signal, and \u03bei<\/sub>(t) is zero-mean noise independent of eij<\/sub>(t). In expectation over the noise, the weight update becomes<\/p>\n

E[\u0394wij<\/sub>(t)] = \u03b7 E[gi<\/sub>(t) eij<\/sub>(t)],<\/p>\n

which still points in a descent (or ascent) direction provided that gi<\/sub>(t) and eij<\/sub>(t) are appropriately correlated with the true gradient. This robustness property explains how coarse neuromodulatory signals or imperfect feedback pathways can still support effective learning, as long as the sign and rough magnitude of the backward signal are statistically informative. In practice, additional regularization terms\u2014such as weight decay or normalization\u2014can be appended to the update rule, yielding<\/p>\n

\u0394wij<\/sub>(t) = \u03b7 bi<\/sub>(t) eij<\/sub>(t) \u2212 \u03b7r<\/sub> \u2202R(w)\/\u2202wij<\/sub>,<\/p>\n

where R(w) is a regularizer enforcing sparsity, stability, or other structural constraints. These extensions are straightforward to integrate within the eligibility-based formalism, and they further align biologically plausible mechanisms with normative optimization principles.<\/p>\n

Finally, retrocausal rules need not be restricted to scalar backward signals; vector- or structure-valued feedback can be incorporated as well. Let Bik<\/sub> denote a feedback matrix mapping errors at an abstract output unit k to neuron i, and let \u03b5k<\/sub>(t) be an output-layer prediction error. The backward signal at neuron i can then be written as<\/p>\n

bi<\/sub>(t) = \u03a3k<\/sub> Bik<\/sub> \u03b5k<\/sub>(t),<\/p>\n

and the synaptic update remains \u0394wij<\/sub>(t) = \u03b7 bi<\/sub>(t) eij<\/sub>(t). This structure underlies algorithms such as feedback alignment and related schemes, where the exact transpose of the forward weights is replaced by a fixed or slowly adapting feedback matrix B. Even when B is not matched to the feedforward connectivity, learning can still converge because the interaction of B with the evolving feedforward weights w indirectly shapes error propagation. In a retrocausal interpretation, this means that backward signals do not need to perfectly mirror the forward pathways; they only need to carry enough structured information so that, when combined with eligibility traces, they guide plasticity in a direction that progressively reduces error or improves reward over time.<\/p>\n

Computational simulations of backward-propagating signals<\/h3>\n

Computational experiments provide a controlled testbed for evaluating how backward-propagating signals and eligibility traces interact to produce effective plasticity. A minimal setup involves a layered or recurrent network in which forward activity is generated in discrete time steps, and a separate backward process delivers evaluative feedback after some delay. At each synapse, a local eligibility variable is updated according to the pre- and postsynaptic activity, while weight changes occur only when a backward signal is present. By systematically varying network architecture, temporal delays, and the form of the backward signals, simulations can reveal which learning rules reliably approximate gradient descent, which merely stabilize behavior, and which fail due to temporal misalignment or noise.<\/p>\n

One common class of simulations uses rate-based neural networks with continuous activation dynamics to approximate the behavior of spiking networks while keeping the analysis tractable. In such models, each neuron integrates weighted inputs and passes them through a nonlinear activation function, and the eligibility trace at a synapse is updated by simple differential or difference equations. The backward signal is typically an error derived from the difference between target and actual outputs, injected at the output layer and sometimes distributed to hidden units via a fixed or random feedback matrix. By comparing performance under these retrocausal update rules to standard backpropagation, researchers can quantify how closely local eligibility-based methods reproduce ideal gradient learning on supervised tasks such as classification, function approximation, or temporal sequence prediction.<\/p>\n

A central parameter in these simulations is the time constant of eligibility traces, which determines how far into the past a backward signal can reach. To explore this, networks can be trained on tasks with controlled input\u2013output delays, such as associating a stimulus at time t with a label or reward at time t + \u0394. When eligibility decays too quickly relative to \u0394, backward signals fail to overlap with the relevant traces, and the network struggles to learn long-range dependencies. Conversely, when eligibility is too long-lived, credit is spread across many irrelevant past events, leading to slow or unstable learning. By sweeping across time constants and measuring learning speed and final accuracy, one can characterize a window of optimal temporal integration where retrocausal rules are both efficient and robust.<\/p>\n

Recurrent neural networks offer a more demanding test because they must solve temporal credit assignment through internal state dynamics as well as through backward signals. In simulations, recurrent units receive continuous streams of inputs and are trained to produce context-dependent outputs, such as remembering a cue across a delay or extracting structure from sequences. Retrocausal learning is implemented by maintaining eligibility traces not only for feedforward synapses but also for recurrent connections, allowing backward signals at a later time to modulate plasticity throughout the loop. Performance can then be compared against recurrent networks trained with backpropagation through time. While eligibility-based retrocausal learning may not achieve the same level of fine-grained optimization, simulations show that it can capture many qualitative features of gradient-trained networks, including stable working memory, selective gating, and task switching, especially when combinatorial regularization and normalization strategies are used.<\/p>\n

Simulations of spiking neural networks provide a complementary perspective by making spike timing an explicit factor in learning. In these models, neurons follow biologically inspired dynamics such as leaky integrate-and-fire or conductance-based equations, and synapses implement STDP-like eligibility updates. A backward signal, often modeled as a neuromodulatory pulse or a structured feedback current, arrives after particular behavioral events such as correct or incorrect responses. When this signal coincides with residual eligibility traces at synapses, those synapses are strengthened or weakened according to a chosen plasticity rule. Tasks such as pattern classification, temporal interval discrimination, and simple decision-making are used to evaluate whether such retrocausal spike-based rules can form stable, generalizable representations. The simulation results typically show that, within realistic ranges of firing rates and neuromodulatory delays, backward-modulated STDP can approximate reinforcement learning algorithms and support nontrivial performance.<\/p>\n

Another important line of computational work examines how backward signals should be routed through multilayer architectures. Direct backpropagation requires using the exact transpose of the forward weights to propagate errors, which is biologically implausible. Simulations of feedback alignment and related mechanisms replace this requirement with random or slowly adapting feedback matrices that guide errors from the output layer back to hidden units. When combined with eligibility traces at hidden-layer synapses, these random feedback signals can nonetheless generate useful learning updates. Studies show that networks trained with such mismatched feedback can approach the performance of backpropagation on a range of tasks, particularly when layer widths are sufficiently large and when regularization stabilizes the evolving representations. These results support the notion that precise symmetry between forward and backward pathways is not necessary; approximate, statistically aligned backward connectivity can suffice when paired with local plasticity and proper credit assignment.<\/p>\n

To evaluate how well retrocausal rules cope with noisy or partial feedback, simulations often introduce stochasticity into the backward signals. For instance, error terms might be corrupted by additive noise, randomly dropped (simulating unreliable neuromodulators), or quantized to a small set of discrete levels. By measuring the resulting degradation in task performance, as well as the stability of weights and firing patterns, one can assess the tolerance of retrocausal learning to imperfect teaching signals. Results generally indicate that, as long as the backward signals retain correct sign information and a rough proportionality to the underlying error, the network can still converge to useful solutions, albeit more slowly. This robustness aligns with theoretical arguments that unbiased but noisy estimates of gradients can sustain learning in stochastic environments.<\/p>\n

Computational experiments also explore how retrocausal mechanisms interact with structural constraints such as sparsity, Dale\u2019s law, and limited precision synapses. For example, one can enforce that each neuron is either excitatory or inhibitory and that backward signals respect these constraints, while eligibility traces remain purely local. Training such constrained networks on sensory discrimination or motor control tasks reveals whether backward-propagating signals can still find effective parameter configurations. Typically, success depends on balancing three factors: sufficient diversity in feedback pathways to distribute evaluative information; enough redundancy in connectivity to accommodate structural constraints; and learning rules that prevent runaway excitation or inhibition. Simulations that satisfy these conditions demonstrate that backward-modulated plasticity can coexist with realistic architectural limitations.<\/p>\n

A particularly informative class of simulations involves environments requiring delayed rewards, making temporal credit assignment especially challenging. Here, agents embodied as retrocausally trained networks interact with a simple environment, choose actions, and receive scalar rewards only after long sequences of decisions. Eligibility traces are maintained at synapses according to the agent\u2019s ongoing activity, and a delayed reward prediction error is delivered as a backward signal when outcomes are revealed. Comparing these agents to those trained with classical reinforcement learning algorithms\u2014such as policy gradients with eligibility traces or Q-learning with function approximation\u2014shows how well retrocausal plasticity can capture long-range dependencies. Results often highlight that performance is strongly shaped by the time constants of eligibility and by the distribution of delays, underscoring the importance of matching synaptic memory to behavioral timescales.<\/p>\n

Another dimension explored in simulations is the interaction between retrocausal learning and ongoing network dynamics such as oscillations, synchrony, and attractor states. Networks with recurrent connectivity may exhibit intrinsic rhythms or metastable patterns even in the absence of external input. Backward signals that arrive at specific phases of these rhythms may interact more or less effectively with eligibility traces, leading to phase-dependent plasticity. Computational studies can vary the timing of backward signals relative to network oscillations and measure how this affects learning speed and stability. These experiments suggest that aligning backward signals with periods of high dendritic excitability or strong recurrent coherence can greatly enhance the impact of retrocausal updates, which may offer an explanation for the behavioral role of event-locked oscillatory bursts observed in biological systems.<\/p>\n

To investigate the relationship between retrocausality and normative optimization, simulations often track not only behavioral performance but also approximate gradients of the loss function or reward objective. By estimating true gradients via backpropagation or finite differences and comparing them to the effective weight changes produced by backward-modulated eligibility traces, one can quantify alignment between the two. Alignment metrics, such as cosine similarity between gradient vectors and actual updates, reveal whether the learning rules move weights in roughly the same direction as ideal gradient descent. Studies show that alignment can be remarkably strong in many regimes, particularly in early training when representations are still flexible. Over time, even moderate alignment is often sufficient for convergence, explaining why relatively crude retrocausal rules can perform well on complex tasks when given adequate network capacity and training data.<\/p>\n

Simulations further explore how priors and prediction-based frameworks interact with backward learning. In generative or predictive coding models, networks attempt to minimize discrepancies between predicted and observed inputs, and backward signals convey prediction errors from higher to lower layers. In this context, computational experiments implement eligibility traces in the generative synapses and use prediction error signals as backward modulators of plasticity. By training such networks on structured sensory input, one can observe the emergence of internal representations that capture latent causes, invariances, and hierarchical structure. Comparisons to equivalent models trained with explicit backpropagation reveal that retrocausal learning rules can approximate variational or bayesian brain\u2013style inference, even though each synapse operates only on local traces and delayed prediction errors.<\/p>\n

An additional question addressed in simulations is how backward-propagating signals interact with offline processes such as replay and consolidation. Model networks can be equipped with mechanisms that periodically reactivate stored activity patterns, either stochastically or under the influence of a separate \u201chippocampal\u201d module. During these reactivation episodes, backward signals based on cumulative reward or error statistics are applied to synapses while eligibility traces are reconstructed from the replayed activity rather than from real-time behavior. Comparing performance with and without replay sheds light on how offline retrocausal updates extend the effective temporal reach of learning beyond the constraints of real-time eligibility decay. In many cases, replay-based retrocausal updates dramatically improve performance on tasks with very long delays or sparse feedback, mirroring the hypothesized role of sleep and rest in biological learning.<\/p>\n

Computational simulations are crucial for mapping the boundaries of stability in retrocausal learning systems. Because backward signals effectively reach into the past of the network\u2019s dynamics, there is a risk of runaway feedback if updates systematically amplify small errors or noise. By scanning over learning rates, feedback gains, trace time constants, and network sizes, one can empirically map out regions of parameter space where learning converges, oscillates, or diverges. Stability analyses using linearization around fixed points or Lyapunov methods can be compared with numerical results to derive practical guidelines for choosing safe operating regimes. These studies show that, with carefully tuned parameters and simple normalization schemes, backward-propagating signals can drive powerful learning without destabilizing the underlying dynamics, paving the way for scalable implementations of retrocausal plasticity in both biological models and artificial hardware.<\/p>\n

Implications for neural coding and artificial networks<\/h3>\n

Allowing signals that effectively flow backward through a circuit reshapes how information is encoded and manipulated at both biological and artificial levels. In neural coding terms, backward-propagating influences mean that the activity of a neuron is no longer determined solely by its immediate inputs and intrinsic dynamics; instead, it is also constrained by future outcomes that feed back as delayed evaluative signals. This creates a bidirectional dependency in which codes are optimized not just for faithful representation of current stimuli, but for their expected relevance to upcoming tasks and rewards. Neurons participating in states that tend to lead to desirable outcomes will be preferentially stabilized by plasticity mechanisms driven by backward signals, skewing the representational landscape toward features that are historically predictive of success.<\/p>\n

From the perspective of population coding, retrocausal learning encourages a form of outcome-gated redundancy. Multiple neurons may initially represent similar aspects of the input space, but backward-modulated plasticity will selectively strengthen those subsets whose activity patterns consistently align with favorable feedback. Over time, distinct functional ensembles form that encode not only sensory or internal variables but also their temporal contingencies with respect to future events. This can yield codes that appear highly task-specific or context-sensitive: the same sensory pattern might be represented differently depending on the anticipated delay to reward or the reliability of subsequent feedback pathways. Such shaping of population codes is difficult to achieve with strictly feedforward, locally supervised learning rules, but emerges naturally when backward signals interact with eligibility traces spanning extended timescales.<\/p>\n

In predictive coding and bayesian brain frameworks, backward signals already play a central role as carriers of predictions and prediction errors. When these signals are permitted to interact directly with synaptic eligibility traces, neural codes become explicitly tuned to minimize future discrepancies between predicted and observed inputs. Synapses encoding priors in a generative model are no longer static repositories of past experience; they are continuously reshaped by backward error streams that reflect current mismatches between expectations and reality. This creates a tight coupling between priors and plasticity: the same top-down prediction that shapes moment-to-moment firing patterns also determines where and when synaptic changes occur, effectively binding representational and learning dynamics into a single coherent process.<\/p>\n

Backward-propagating signals also blur the traditional divide between \u201ccoding\u201d and \u201ccontrol\u201d in neural systems. In classical views, early sensory areas encode information, while higher areas read out these codes to generate decisions and actions. With retrocausal influences, higher areas not only decode but also sculpt the lower-level codes by delivering outcome-dependent feedback. This means that what counts as a good code is defined in part by its downstream utility for control and decision-making. As a result, early sensory representations may embody implicit control-theoretic quantities, such as expected value or future information gain, embedded within what would otherwise be interpreted as purely sensory tuning curves. Selective backward plasticity can thus explain why neural responses often mix sensory, motor, and cognitive variables in seemingly heterogeneous ways.<\/p>\n

At the level of temporal coding, backward learning encourages representations that are predictive rather than merely reflective of recent input. Neurons that fire in advance of critical events and whose activity reliably forecasts upcoming outcomes are preferentially reinforced by delayed feedback. This can lead to the emergence of ramping activity, anticipatory firing, and phase-precession-like phenomena, where temporal sequences encode the expected trajectory toward future states. Because plasticity is gated by backward signals arriving after these predictive patterns occur, the system effectively performs a form of temporal difference learning on its own population codes, stabilizing those dynamics that best anticipate the structure of future rewards, errors, or surprises.<\/p>\n

Backward influences are equally transformative for artificial neural networks. Standard backpropagation already uses an algorithmic backward pass to propagate errors and compute gradients, but this process is typically dissociated from the network\u2019s runtime coding properties. Introducing biologically inspired, eligibility-based backward learning rules changes this picture. Hidden representations are no longer shaped only by instantaneous gradients tied to a single batch of data; instead, they are refined by sequences of backward events that may be delayed, noisy, and structurally mismatched with the forward pathways. This tends to induce representations that are more robust to temporal variations, resilient to feedback perturbations, and better aligned with the statistics of delayed supervision, such as sparse labels or infrequent rewards.<\/p>\n

In deep architectures, backward-propagating signals that do not precisely mirror the forward weights push the network toward internal codes that are inherently error-correcting. Because feedback pathways may be random or only approximately aligned with the forward connections, hidden units that develop representations strongly correlated with true task-relevant factors are the ones that can reliably extract useful teaching information from noisy backward streams. This selection pressure can lead to overcomplete, distributed codes that tolerate feedback distortions. Empirical studies with feedback-alignment-like algorithms show that, even without symmetric weights, networks can learn hierarchical features resembling those obtained with backpropagation, suggesting that retrocausal constraints still allow for the emergence of deep, compositional structure under realistic routing of backward signals.<\/p>\n

Recurrent and reservoir-style networks particularly benefit from backward-modulated plasticity, as their coding power depends heavily on how internal dynamics carve out stable and metastable attractor states. Retrocausal updates encourage attractors that are not only internally self-consistent but also behaviorally useful: patterns that tend to precede positive outcomes are deepened as basins in state space, while those that predict errors are shallowed or eliminated. This reshaping of the dynamical landscape alters the network\u2019s temporal codes, making it more likely to converge onto trajectories that historically led to favorable backward feedback. Artificial agents trained in this way may develop internal \u201cschemas\u201d or \u201ctask sets\u201d represented as robust dynamical modes, facilitating rapid adaptation when tasks reoccur or when feedback statistics change gradually over time.<\/p>\n

Retrocausal mechanisms also suggest new ways of organizing memory in artificial systems. Instead of treating memory as a passive buffer that stores past activations for later gradient computation, networks can maintain active eligibility traces that interact with backward signals over extended intervals. This supports learning from temporally sparse data streams, where explicit replay or full unfolding through time would be computationally expensive. By tuning the decay of eligibility and the schedule of backward updates, artificial systems can emulate the multi-timescale memory structures observed in biology: fast, labile traces for recent events coexisting with slower, more persistent traces that accumulate statistics over many episodes. Neural codes in such systems become stratified across timescales, with certain units specializing in short-latency predictions and others encoding more abstract, long-horizon contingencies.<\/p>\n

The interplay between backward signals and normalization or homeostatic mechanisms is another key determinant of coding in artificial networks. Because retrocausal plasticity can amplify or suppress specific pathways based on delayed evaluation, there is a risk that a few highly rewarded patterns will dominate, leading to brittle, overfitted representations. Incorporating synaptic normalization, activity regularization, or constraint-enforcing projections tempers this effect, forcing the network to distribute credit across multiple coding dimensions. The resulting representations are often sparser yet more diverse, balancing specialization with coverage of the input space. This aligns with experimental observations of sparse, overcomplete coding in cortex and provides a design principle for artificial architectures that aim to exploit backward learning without sacrificing generalization.<\/p>\n

Backward-propagating influences also reshape how artificial networks handle uncertainty. When evaluative signals reflect not only mean performance but also variability, synapses may be adjusted to encode confidence or precision alongside expected value. For instance, backward signals derived from probabilistic objectives can modulate plasticity in proportion to the estimated reliability of an error, causing representations to sharpen for consistently informative features while remaining diffuse for ambiguous or noisy inputs. This effectively embeds a rudimentary measure of precision into the network\u2019s codes, echoing bayesian brain theories in which higher-level circuits encode both expectations and their associated uncertainties. In practice, such coding can support more robust decision-making in the face of distributional shifts or partial observability.<\/p>\n

In architectures that integrate symbolic or modular components, retrocausal learning offers a principled mechanism for assigning credit across modules. When a composite system performs a sequence of transformations, backward signals arriving at later stages can propagate into earlier modules in a way that depends on the compatibility between their internal codes. Modules whose representations align well with the \u201clanguage\u201d of the backward signal will receive clearer learning guidance and thus adapt more quickly. Over time, this can drive the emergence of common representational interfaces across modules, facilitating communication and reuse. Such alignment-by-backward-plasticity may be a route toward more interoperable, compositional AI systems without requiring explicit hand-engineering of shared symbol sets or communication protocols.<\/p>\n

Backward influences highlight the importance of temporal organization in training regimes for artificial networks. Because the efficacy of retrocausal updates depends on the overlap between eligibility traces and backward signals, the order and pacing of experiences matter. Interleaving tasks, shaping reward schedules, or structuring curricula can alter which internal codes are stabilized and which remain plastic. For example, presenting easier, highly feedback-informative tasks early on can seed a set of robust feature detectors and temporal predictors that later tasks can build upon, even if their feedback is sparser or more delayed. This mirrors educational strategies and developmental trajectories in biological organisms, where early, dense feedback sculpts foundational codes that bias subsequent learning in favorable directions.<\/p>\n","protected":false},"excerpt":{"rendered":"

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