# A Unified Covariance and Spike-Timing Dependent Hebbian Plasticity Model (CovSTDP) for Neural Adaptation

Kamal Al-Malah

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Kamal Al-Malah. A Unified Covariance and Spike-Timing Dependent Hebbian Plasticity Model (Cov-STDP) for Neural Adaptation. 2025. hal-05412507

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# A Unified Covariance and Spike-Timing Dependent Hebbian Plasticity Model (CovSTDP) for Neural Adaptation

Author: Kamal Al-Malah, Ph.D.

Book Author and Consultant, Irbid, JORDAN

k.almalah@gmail.com

## Abstract

Synaptic plasticity models describe how pre-synaptic and post-synaptic activity drives learning in biological neural systems, making them essential for studying memory formation and adaptive behavior in the brain. This work presents a unified MATLAB framework implementing six major Hebbian-type plasticity mechanisms, Hebbian, Covariance, BCM, Oja, STDP, and a hybrid CovSTDP rule, within a wake-sleep learning architecture. The code provides a clean, reproducible platform that isolates the contribution of each plasticity rule to network learning dynamics.

The models were evaluated using several categories of synthetic data, ranging from simple linearly separable Gaussian clusters to temporally perturbed and more irregular patterns. Performance was assessed through classification accuracy, loss evolution, and synaptic energy growth. Results show that while all six models perform well on simple datasets, differences emerge under more complex inputs, with the CovSTDP hybrid consistently achieving the highest accuracy and most stable convergence.

Overall, the study demonstrates that plasticity rules combining correlation-based and timing-dependent components offer superior computational and biological fidelity, making CovSTDP the strongest candidate among the evaluated mechanisms.

## Introduction

Understanding how biological and artificial systems learn has become a central theme in modern neuroscience and artificial intelligence research [1,2]. Although artificial neural networks (ANNs) and the human brain differ fundamentally in their physical implementation, both systems exhibit striking similarities in how they adapt internal connections to improve performance over time [2]. These similarities are often described as *computational parallels*, meaning that distinct physical systems can perform analogous
mathematical or logical operations using different mechanisms [1]. In this context,
computational parallels emerge when comparing algorithmic learning in ANNs with
synaptic plasticity and memory processes in biological neural circuits, particularly during
learning and sleep [3,4,5].

One of the most influential learning mechanisms in artificial intelligence is backpropagation (backward propagation of errors) [1]. Backpropagation enables ANNs to learn from experience by iteratively reducing the error between predicted and desired outputs [1,2]. In this process, a network first generates an output, then computes the discrepancy between the predicted and true values, and finally propagates this error backward through the network to update synaptic weights. Mathematically, this learning framework is grounded in gradient descent, which adjusts parameters in the direction that most efficiently minimizes the loss function [1]. Conceptually, backpropagation allows artificial systems to “learn from their mistakes” by systematically refining internal representations over successive iterations [2].

In biological systems, learning follows a related but biologically grounded process known
as synaptic consolidation [3,4,5]. Unlike the instantaneous weight changes in artificial
networks, synaptic consolidation occurs over extended periods and is especially
prominent during sleep or quiet wakefulness [3,4]. During active learning, neurons
establish transient and labile connections that encode short-term memories. Subsequently,
during sleep, neural circuits replay recent activity patterns, selectively strengthening
important synapses while weakening redundant or noisy ones [3,4,5]. This process
enables the gradual transfer of memory traces from short-term storage in the
hippocampus to long-term storage in the cortex [3,4]. From a computational perspective,
synaptic consolidation closely resembles an offline optimization stage, in which
previously acquired data are revisited to refine internal connectivity without the need for
continuous external input [4,5].

The conceptual alignment between backpropagation and synaptic consolidation illustrates a powerful bridge between neuroscience and machine learning [6,2]. While backpropagation operates through explicit gradient calculations and global error signals, synaptic consolidation relies on distributed, local activity-dependent mechanisms shaped by sleep-related neural replay [3,4,6]. Nevertheless, both processes achieve a common functional outcome: the stabilization of useful patterns and the suppression of ineffective or noisy connections [6,2]. Table 1 summarizes these conceptual parallels, presenting simplified interpretations and direct analogies that highlight how biological learning principles can inform the design of more energy-efficient and adaptive computational models [6,2].
<table><caption>Table 1: Summary of conceptual parallels between computational and biological learning processes, illustrating simplified meanings and analogies across domains.</caption><thead><tr><td>Concept</td><td>Domain</td><td>Simplified Meaning</td><td>Analogy</td></tr></thead><tbody><tr><td>Computational Parallel</td><td>General</td><td>Two systems performing similar operations differently</td><td>Brain ≈ Computer</td></tr><tr><td>Backpropagation</td><td>Artificial Intelligence</td><td>Adjusting connection strengths to reduce error</td><td>Learning from mistakes</td></tr><tr><td>Synaptic Consolidation</td><td>Neuroscience</td><td>Strengthening or weakening brain connections during rest</td><td>Saving important lessons to long-term memory</td></tr></tbody></table>

This foundational understanding provides the conceptual groundwork for exploring biologically inspired learning rules, including hybrid Hebbian models that integrate correlation-based and timing-based plasticity [6,2], as well as motivating sleep-like phases in artificial learning systems [3,4,5]. By bridging artificial learning algorithms with biological memory mechanisms, new opportunities emerge for the development of robust, low-energy, and self-organizing intelligent systems [6,2]. From a computational perspective, these biological processes resemble an offline update phase in which a network revisits previously acquired information to refine internal connections without continuous sensory input [4,5]. This analogy highlights strong conceptual parallels between computational and biological learning, as summarized in Table 1, which presents simplified meanings and cross-domain analogies. Together, these insights prepare the foundation for deeper investigation into how sleep-based learning mechanisms can inspire more energy-efficient and biologically plausible computational models [2-6].
# Theory

## 1. Hebbian Learning Rule [7]

Hebb's central goal was to bridge the gap between psychology (the study of the mind and behavior) and physiology (the study of the brain and nervous system). He proposed a concrete, physiological theory for how mental processes, like learning, memory, and perception, could arise from the brain's activity. His most famous and enduring contribution from this work is "Hebb's Rule". It is often summarized as "cells that fire together, wire together." Hebb proposed that when one neuron persistently participates in firing another, the connection between them becomes stronger and more efficient, a process that provides a biological mechanism for learning and memory formation. Through this rule, Hebb explained how neural connections are reinforced and stabilized, making it possible for experiences to leave lasting imprints in the brain's structure. Building on this principle, he introduced the concept of the cell assembly, a network of neurons that become strongly interconnected through repeated co-activation. Each cell assembly functions as a single unit representing an idea or perception; for instance, one's mental representation of an "apple" arises from the activation of a specific neural assembly dedicated to that concept.

To explain how thoughts flow coherently rather than appearing as isolated events, Hebb proposed the idea of phase sequences, in which one cell assembly's activation leads to the activation of another, forming a sequence of related thoughts. This dynamic process underlies reasoning, planned behavior, and conscious mental activity. Finally, Hebb introduced the concept of reverberatory circuits to describe how neural activity could persist briefly without continuous external stimulation. These self-sustaining loops of activity allow a cell assembly to remain active for a short duration, providing a physiological explanation for temporary memory and thought maintenance. Collectively, these concepts laid the groundwork for understanding the neural basis of learning, memory, and cognition.

Mathematically, the change in connection strength can be expressed [2,3] as:

$$ \Delta w_{ij} = \eta x_i y_j \quad (1) $$

where $x_i$ and $y_j$ are the firing levels of neurons i and j, and $\eta$ is the learning rate.

Eq. (1) can lead to runaway growth if unregulated,

## 2. Covariance Hebbian Rule [8]

An alternative form [4], called covariance Hebbian can be used:

$$ \Delta w_{ij} = \eta(x_i - \bar{x})(y_j - \bar{y}) \quad (2) $$
Eq. (2) allows for weight decreases if one neuron is active while the other is below its average activity.

Note: The variable names are sometimes swapped ($y_i$ for postsynaptic, $x_j$ for presynaptic), but the mathematical form is identical to the above equations. Alternatively, $x_i$ is the sending neuron's activation, $y_j$ is the receiving neuron's activation.

### 3. The Bienenstock-Cooper-Munro (BCM) Rule [9]

BCM introduces a dynamic threshold for postsynaptic activity, strengthening connections when firing exceeds an adaptive mean level and weakening them when below it, thus maintaining homeostatic stability.

$$ \Delta w_{ij} = \eta y_j (y_j - \theta_M) x_i \quad (3) $$

where

$x_i$: presynaptic activity (input neuron $i$)

$y_j$: postsynaptic activity (output neuron $j$)

$\theta_M$: sliding modification threshold, typically the long-term average of $y_j^2$

$\eta$: learning rate

Eq. (3) implies that:

* If the neuron fires much higher than its average ($y > \theta_M$) → it strengthens the connection denoted as long-term potentiation (LTP).

* If it fires less than its average ($y < \theta_M$) → it weakens what is called the long-term depression (LTD).

* The threshold $\theta_M$ moves dynamically, making the neuron self-stabilizing and avoiding runaway excitation.

### 4. Oja's Rule [10]

Oja's rule refines Hebbian learning by incorporating a normalization term that prevents unbounded weight growth, effectively ensuring that learned representations remain stable while still reinforcing correlated activity. The rule is given by:

$$ \Delta w_{ij} = \eta y_j (x_i - y_j w_{ij}) \quad (4) $$

where

$x_i$: presynaptic activity (input neuron $i$)
$y_j$: postsynaptic activity (output neuron j)

$w_{ij}$: weight from neuron i → j

$\eta$: learning rate

Eq. (4) implies that:

* The first term ($y_j x_i$) strengthens weights when pre- and post-synaptic neurons are co-active (classic Hebb).

* The second term ($-y_{ij}^2 w_i$) prevents unbounded growth by normalizing the total weight energy.

So, Oja's rule essentially performs Hebbian learning with automatic normalization, making it a simple model of how neurons might keep their activity bounded while learning correlations.

## 5. Spike-Timing Dependent Plasticity (STDP) [11]:

STDP emphasizes the temporal order of neural firing: If a presynaptic spike precedes a postsynaptic one within a narrow time window, the synapse strengthens (causal learning), whereas reversed timing leads to weakening. Together, these rules illustrate progressively more realistic refinements of Hebbian learning, embedding correlation, normalization, and temporal causality into the adaptive fabric of biological neural networks.

The change in synaptic weight depends on the timing difference between pre- and post-synaptic spikes. The rule is given by:

$$ \Delta w_{ij} = \begin{cases} A_+ e^{-\frac{\Delta t}{\tau_+}}, & \text{if } \Delta t = t_{post} - t_{pre} > 0 \\ -A_- e^{-\frac{\Delta t}{\tau_-}}, & \text{if } \Delta t = t_{post} - t_{pre} < 0 \end{cases} \qquad (5) $$

where

* $\Delta w_{ij}$: change in synaptic weight from presynaptic neuron i to postsynaptic neuron j

* $\Delta t$: timing difference between post- and pre-synaptic spikes

* $A_+$ → amplitude of long-term potentiation (LTP)

* $A_-$ → amplitude of long-term depression (LTD)

* $\tau_+$ and $\tau_-$ → corresponding time constants for decay of each effect

* The "+" and "-" on $A_+$ and $A_-$ are *superscripts* (not arithmetic signs)
Notice that in rate-based Hebbian learning, the rule is written in terms of continuous neural activities: $\Delta w_{ij} = \eta f(x_i, y_j)$

where $x_i$ and $y_j$ are typically firing rates (averaged over time). However, STDP works at the spike level, not at averaged rates.

In STDP, the key variable is the exact timing between individual pre- and post-synaptic spikes: $\Delta t = t_{post} - t_{pre}$. The weight change is then determined solely by this timing difference, not by mean firing rates.

One can think of STDP as a discretized, event-driven version of Hebbian learning:

* When a pre-synaptic spike occurs shortly before a post-synaptic spike, the connection is strengthened (long-term potentiation, LTP).

* When a pre-synaptic spike follows a post-synaptic spike, the connection is weakened (long-term depression, LTD).

From a biological insight, STDP captures the causal flow of information: Neurons that predict firing of others get rewarded.

## 6. A Proposed Hybrid Covariance-STDP Learning Rule (CovSTDP)

An attempt is to combine STDP's temporal selectivity with covariance Hebbian's sensitivity to deviations from baseline so that only meaningful, timely pre-post correlations (those that differ from average activity) produce durable weight changes. This rule is formulated as a spike-pair (event-based) STDP mechanism modulated by activity covariance.

Let $t_{pre}$ and $t_{post}$ be spike times of pre- and post-synaptic neurons. Define the usual STDP kernel:

$$K(\Delta t) = \begin{cases} A_+ e^{-\frac{\Delta t}{\tau}}, & \text{if } \Delta t = t_{post} - t_{pre} > 0 \\ -A_- e^{-\frac{\Delta t}{\tau}}, & \text{if } \Delta t = t_{post} - t_{pre} < 0 \end{cases} \qquad (6)$$

Define the covariance gate for the same event pair using short-term running means $\mu_i, \mu_j$ or low-pass filtered activities:

$$C_{ij}(t) = (x_i(t) - \mu_i(t))(y_j(t) - \mu_j(t)) \qquad (7)$$

Then the hybrid weight change for that spike pair is:

$$\Delta w_{ij} = \eta C_{ij}(t) K(\Delta t) \qquad (8)$$
Eq. (8) says: An STDP timing contribution $K(\Delta t)$ is scaled up/down by whether the pre/post activities are above or below their recent means. If $C_{ij}$ is near zero (no unusual covariance), timing will have a little effect.

Alternatively, if we have continuous rate signals $x_i(t)$, $y_j(t)$ (or spike-rate traces), we can average over many pairs and get:

$$ \Delta w_{ij}(t) \approx \eta \langle (x_i(t) - \mu_i(t))(y_j(t) - \mu_j(t)) * K(\tau) \rangle_{\tau} \quad (9) $$

Eq. (9) says that in the continuous or rate-based approximation, the discrete spike events are replaced by smooth firing-rate signals $x_i(t)$ and $y_j(t)$, allowing the learning rule to be expressed in terms of averaged correlations rather than individual spikes. Here, the synaptic weight change $\Delta w_{ij}(t)$ reflects the covariance between pre- and post-synaptic rates, temporally filtered by the STDP kernel $K(\tau)$, meaning that the timing of rate fluctuations still influences learning even without explicit spikes.

In conclusion, the proposed rule integrates covariance-based correlation detection with spike-timing-dependent modulation, yielding a hybrid Hebbian-like formulation. Unlike purely covariance or timing-based models, the CovSTDP rule strengthens synapses when presynaptic and postsynaptic activities are both above their mean levels and temporally aligned in a causal order. Mathematically, the weight change is expressed as

$$ \Delta w_{ij} = \eta F(\Delta t) (x_i - \langle x_i \rangle)(y_j - \langle y_j \rangle) \quad (10) $$

where $F(\Delta t)$ represents an asymmetric STDP kernel that decays exponentially with the time difference between post- and presynaptic activations. This hybrid formulation allows the network to capture both the statistical dependencies of continuous-rate signals and the temporal precision of spike-based interactions, potentially improving stability and selectivity under noisy or dynamic input conditions.
Table 1 shows a comparison of six Hebbian-type learning mechanisms, highlighting the proposed hybrid CovSTDP rule, which merges covariance-based and timing-dependent dynamics to achieve biologically plausible and stable synaptic adaptation.

<table><caption>**Table 2:** Comparison of six Hebbian-type learning mechanisms from different perspectives.</caption><thead><tr><td>Learning Rule</td><td>Stability</td><td>Biological Plausibility</td><td>Temporal Sensitivity</td><td>Computational Complexity</td><td>Interpretation</td></tr></thead><tbody><tr><td>Simple Hebbian</td><td>Unstable; can diverge without normalization</td><td>Simplistic correlation rule</td><td>None (instantaneous)</td><td>Very low</td><td>Raw co-activity</td></tr><tr><td>Covariance Hebbian</td><td>Mean-centering adds intrinsic stability</td><td>Closer to cortical Hebbian plasticity</td><td>None</td><td>Requires running averages</td><td>Correlation beyond chance level</td></tr><tr><td>BCM</td><td>Stabilized by adaptive threshold (θ<sub>M</sub>)</td><td>Captures activity-dependent thresholding</td><td>Low (activity average)</td><td>Moderate</td><td>Homeostatic rate control</td></tr><tr><td>Oja</td><td>Normalized via output self-limiting term</td><td>Models firing-rate homeostasis</td><td>Low</td><td>Moderate</td><td>Normalized Hebbian learning</td></tr><tr><td>STDP</td><td>Stable if balance between potentiation and depression maintained</td><td>Realistic timing-based plasticity</td><td>High (millisecond-scale)</td><td>Higher (time-dependent kernel)</td><td>Causal timing modulation</td></tr><tr><td>Hybrid CovSTDP (proposed)</td><td>Stable via joint temporal-covariance</td><td>Integrates both rate- and spike-based</td><td>Moderate-high — timing weighted by covariance</td><td>Moderate-high (adds temporal kernel to covariance)</td><td>Correlation gated by spike timing (causal covariance)</td></tr></tbody></table>
<table><thead><tr><td>Learning Rule</td><td>Stability</td><td>Biological Plausibility</td><td>Temporal Sensitivity</td><td>Computational Complexity</td><td>Interpretation</td></tr></thead><tbody><tr><td></td><td>normalization</td><td>plasticity</td><td>strength</td><td></td><td></td></tr></tbody></table>

In artificial intelligence, Hebbian rules inspired early neural networks and remained foundational in energy-efficient or unsupervised learning models. [1]

Analogy: Two musicians who practice together repeatedly improve their coordination; their connection strengthens naturally.

# Methodology

## 1. Algorithmic Summary of the Wake-Sleep Neural Learning Simulation

This algorithm describes the sequential logic implemented in the MATLAB wake-sleep learning simulation. The model emulates biologically inspired learning by alternating between wake (online synaptic plasticity) and sleep (memory replay and consolidation) phases, while supporting multiple Hebbian-style plasticity rules and optional supervised refinement.

### 1. Initialization phase

All network parameters are initialized, including synaptic weight matrices (W1, W2), bias terms (b1, b2), learning rates, decay regularization, and random seed. If no external data are provided, the code generates temporal 2-D input signals (sine and cosine waves with noise), rather than static Gaussian clusters, and constructs a delayed binary target sequence for sequence learning.

### 2. Dataset and parameter validation

The code validates input dimensions, assigns default training options, and initializes running averages used by covariance- and BCM-type learning rules.

### 3. Epoch-wise training loop (Wake-Sleep cycle)

For each training epoch, we have:

Wake phase (online learning):

a) A single random sample from the temporal sequence is selected.

b) Hidden-layer activations are computed using a tanh nonlinearity and normalized.

c) Output activation is computed using a sigmoid function.
d) Synaptic updates are applied according to the selected biological learning rule (Hebbian, Covariance, BCM, Oja, STDP, or hybrid CovSTDP).

e) If enabled, a supervised wake refinement step applies a small gradient-based correction using the target label.

Sleep phase (memory replay):

a) A small batch of patterns is randomly replayed.

b) Noise is injected into hidden activations to simulate memory reactivation.

c) A supervised consolidation step (sigmoid + backpropagation-style update) adjusts the weights using a lower learning rate.

4. Monitoring and logging

After each epoch, the algorithm computes and stores the mean-squared error loss and a synaptic energy measure (sum of squared weights). Periodic progress messages are printed during training.

5. Post-training evaluation

After training finishes, the network:

a) Computes sigmoid-based output probabilities.

b) Converts them to binary class labels using a fixed or adaptive threshold.

c) Computes the confusion matrix and class-wise as well as overall accuracy.

d) Stores all results and final parameters in the history structure.

Unlike simple static demonstrations, this implementation emphasizes temporal sequence learning, biologically inspired plasticity rules, stochastic neural drift, and explicit sleep-phase consolidation.

## 2. Explanation of Input Variables ($x_1$ and $x_2$) (Revised)

In this implementation, $x_1$ and $x_2$ do not represent fixed Gaussian-cluster features. Instead, they are the two components of a time-varying temporal input signal.

Each training sample is a two-dimensional vector:

$$x(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}$$

where:

* $x_1(t)$ is a noisy sine-wave signal.

* $x_2(t)$ is a noisy cosine-wave signal.

Both signals are normalized and further modulated by a slowly varying temporal component to introduce non-stationarity.
The target output is not based on spatial class clusters, but on temporal dependency:

* The desired label at time *t* depends on a delayed version of $x_1(t)$.

* The task is therefore a sequence prediction / temporal classification problem, where the network learns to associate current inputs with past signal states, rather than separating static spatial clusters.

In summary, $x_1$ and $x_2$ are dynamic time-series inputs, and the learning objective is temporal pattern association, not spatial Gaussian cluster classification.

### 3. Purpose of the MATLAB Code

The primary purpose of this MATLAB code is to simulate the learning dynamics of a biologically inspired wake-sleep neural network operating on temporal input signals. Rather than classifying static Gaussian clusters, the model learns to associate time-varying two-dimensional sensory inputs with a binary target sequence that depends on delayed signal structure.

The code is designed to:

a) Emulate biologically plausible learning through alternating wake (online, local plasticity) and sleep (memory replay and consolidation) phases.

b) Compare and study multiple Hebbian-style learning rules (Hebbian, Covariance, BCM, Oja, STDP, and a hybrid Covariance-STDP rule).

c) Investigate how stochastic neural drift, synaptic decay, and sleep-phase replay affect temporal prediction and classification performance.

In essence, the simulation provides a computational framework for studying sequence learning, biologically motivated plasticity, and memory consolidation, rather than simple static pattern separation.

### 4. Interpretation of Loss and Energy Evolution

During training, the model continuously tracks two key quantities: loss and energy, which provide complementary insights into the learning dynamics.

#### 1. Loss (Prediction Error)

The loss is computed as the mean squared error (MSE) between the network's sigmoid output and the target binary sequence. As training proceeds, the loss generally decreases as the network improves its ability to predict temporally delayed target states from the current inputs. Oscillations in the loss are expected due to the stochastic sampling of inputs, injected noise, and biologically inspired plasticity rules. A long-term downward trend in loss indicates successful acquisition of temporal associations.
## 2. Energy (Synaptic Strength Measure)

The energy is defined as the sum of squared synaptic weights (||$W_1$||² + ||$W_2$||²).

This quantity reflects the overall magnitude of synaptic connectivity in the network. Throughout learning, energy may increase, stabilize, or fluctuate due to the competition between synaptic growth (Hebbian and STDP-like potentiation) and regularization (weight decay). Biologically, this behavior models the balance between synaptic strengthening during learning and homeostatic stabilization, analogous to how natural neural systems regulate plasticity to avoid runaway excitation.

Together, the evolution of decreasing loss and regulated energy illustrates a biologically meaningful trade-off between accuracy of temporal prediction and metabolic/synaptic efficiency, highlighting the efficiency-stability balance observed in natural intelligence.

# 5. MATLAB Code for Testing the Six Hebbian-like Models

Figure 1 shows the MATLAB Code for both the main calling script and the called function which evaluates the performance of the six Hebbian-like plasticity models.

### %% compare_learning_rules.m

### % Compare performance of Hebbian-type learning rules using the wake-sleep model

clear; clc;

base_OPTS = struct('n_epochs', 300, ...

'n_hidden', 60, ...

'eta_wake', 1e-2, ...

'eta_sleep', 8e-3, ...

% increased slightly

'verbose', false, ...

'noGraphics', true, ...

'seed', 1, ...

'supervisedWake', true, ...

'eta.super_wake', 5e-3, ...

% reduced for better stability

'alpha', 0.95, ...

% slower running-average decay

'useAdaptiveThresh', true);

rules = {'Hebbian', 'Covariance', 'BCM', 'Oja', 'STDP', 'CovSTDP'};

results = struct();

for i = 1:length rules)

fprintf('\n===== Running rule: %s =====\n', rules{i});

opts = base_OPTS;

opts.ruleType = rules{i};
% Train and evaluate
history = wake_sleep_network_v2I([], [], opts);
% Store key performance metrics
results.(rules{i}).finalLoss = history_finalLoss;
results.(rules{i}).finalEnergy = history_finalEnergy;
results.(rules{i}).accuracy = history accu racy;
end

%% --- Display summarized table ---
fprintf('\nSummary of Model Performance:\n');
fprintf('-------------------------------------------\n');
fprintf('%-12s %-12s %-12s %-12s\n', 'Rule', 'Loss', 'Energy', 'OverallAcc(%)');
fprintf('-------------------------------------------\n');

for i = 1:length rules)
    rule = rules{i};
    fprintf('%-12s %-12.4f %-12.4f %-12.2f\n', ...
        rule, ...
        results.(rule).finalLoss, ...
        results.(rule).finalEnergy, ...
        100 * results.(rule).accuracy.overall);
end
fprintf('-------------------------------------------\n');

%% Optional: visualize comparison
figure('Name','Learning Rule Comparison','NumberTitle','off');
bar(categorical rules), cellfun(@r)100*results.(r).accuracy.overall, rules));
ylabel('Overall Accuracy (%)');
title('Performance Comparison of Hebbian-Type Learning Rules');
grid on;

function history = wake_sleep_network_v2I(X, Y_target, opts)
% Supports six Hebbian-like rules: Hebbian, Covariance, BCM, Oja, STDP, CovSTDP
%
%% --- Built-in Demo (if no inputs) ---
%% --- Built-in Demo (if no inputs) ---
if margin == 0 || isempty(X) || isempty(Y_target)
% ======== Temporal (sequence-based) demo data for STDP / CovSTDP ========
T = 400;
% total time steps
```c
f = 0.05;      % signal frequency
delay = 5;     % temporal delay (samples)
noise_level = 0.1;
t = 1:T;

% Create 2D temporal signal
Xsig1 = sin(2*pi*f*t) + noise_level*randn(1,T);
Xsig2 = cos(2*pi*f*t) + noise_level*randn(1,T);
X = [Xsig1; Xsig2];

% Normalize
X = X ./ max(abs(X), [], 'all');

% Target depends on delayed version of the signal
Y_target = double([zeros(1,delay), Xsig1(1:end-delay)] > 0);
temporal_mod = 0.3 * sin(t);
X = X + temporal_mod;

% Default training options for demonstration
opts.figPath = './figures/';
opts.n_hidden = 40;
opts.n_epochs = 120;

% opts.seed = randi(10000); % random seed per run
if ~isfield(options, 'seed') || isempty(options.seed)
    options.seed = 1; % fixed default seed for reproducibility
end

% fprintf('Running built-in stochastic Gaussian demo (%d samples per class)
', n_per_class);
fprintf('Running built-in stochastic Gaussian demo
');

%% --- Defaults ---
if length < 3 || isempty(options), options = struct(); end
defaults = struct('n_hidden',15,'eta_wake',5e-3,'eta_sleep',2e-3,'lambda',1e-4,...
    'n_epochs',100,'batch_size',10,'verbose',true,'seed',1,'ruleType','Hebbian');
% defaults = struct('n_hidden',15,'eta_wake',2e-3,'eta_sleep',1e-3,'lambda',5e-4,...
% 'n_epochs',100,'batch_size',10,'verbose',true,'seed',1,'ruleType','Hebbian');
f = fieldnames(defaults);
for k = 1:numel(f)
    if ~isfield(options,{f{k}}) || isempty(options.(f{k}))
        options.(f{k}) = defaults.(f{k});
    end
end

rng(options.seed);
```
%% --- Init ---

[nInputs, nPatterns] = size(X);
n_hidden = opts.n_hidden;
W1 = 0.1*randn(n_hidden,nInputs);
W2 = 0.1*randn(1,n_hidden);
b1 = zeros(n_hidden,1);
b2 = 0.1*randn(1,1);
b2 = b2 + 0.05*randn(); % small stochastic bias each epoch
mean_x = zeros(nInputs,1);
mean_a1 = zeros(n_hidden,1);
mean_y = 0;
history loss = zeros optsn Epochs, 1;
history energy = zeros optsn Epochs, 1;
%% --- Train ---
for epoch = 1:opts.n_epochs
    idx = randi(n_patterns);
    x = X(:,idx);
    a1 = tanh(W1*x + b1); % hidden activations
    a1 = a1 / (norm(a1) + 1e-6);
    b2 = b2 + 0.01 * randn(); % small stochastic bias drift
    sigmoid = @ (z) 1/(1+exp(-z));
    y_raw = W2 * a1 + b2; % pre-activation (1 x 1)
    y_pred = sigmoid(y_raw); % scalar in (0,1)
    alpha = opts.alpha;
    mean_x = alpha*mean_x + (1-alpha)*x;
    mean_a1 = alpha*mean_a1 + (1-alpha)*a1;
    mean_y = alpha*mean_y + (1-alpha)*y_pred;
    % --- Learning rule switch ---
    % --- Learning rule switch ---
    switch lower optsn ruleType
        case 'hebbian'
            W1 = W1 + optsn.eta_wake*(a1*x') - optsn.eta_wake*optsn(lambda*W1;
            W2 = W2 + optsn.eta_wake*(y_pred*a1') - optsn.eta_wake*optsn(lambda*W2;
        case 'covariance'
            W1 = W1 + optsn.eta_wake*((a1-mean_a1)*(x-mean_x)) -
                optsn.eta_wake*optsn(lambda*W1;
            W2 = W2 + optsn.eta_wake*((y_pred-mean_y)*(a1-mean_a1)) -
                optsn.eta_wake*optsn(lambda*W2;
        case 'bcm'
            % Add the remaining lines here
theta = mean(a1.^2);

deltaW1 = (a1.(a1 - theta)).x';

deltaW2 = (y_pred*(a1 - theta))';

W1 = W1 + opts.eta_wake*deltaW1 - opts.eta_wake*opts(lambda)*W1;

W2 = W2 + opts.eta_wake*deltaW2 - opts.eta_wake*opts(lambda)*W2;

case 'oja'

deltaW1 = a1*x' - diag(a1.^2)*W1;

deltaW2 = (y_pred*a1') - (y_pred^2)*W2;

W1 = W1 + opts.eta_wake*deltaW1 - opts.eta_wake*opts(lambda)*W1;

W2 = W2 + opts.eta_wake*deltaW2 - opts.eta_wake*opts(lambda)*W2;

case 'stdp'

tau = 0.1; dt = randn(size(a1)).tau;

W1 = W1 + opts.eta_wake*(exp(-abs(dt)/tau).*(a1*x')) -

opts.eta_wake*opts(lambda)*W1;

W2 = W2 + opts.eta_wake*(exp(-abs(mean(dt))/tau).*(y_pred*a1')) -

opts.eta_wake*opts(lambda)*W2;

case 'covstdp' % <-- NEW hybrid rule

% ---- Tunable parameters ----

tau_trace = 0.2;

A_PLUS = 1.0;

A_minus = 0.7;

% ---- Eligibility traces (activity-dependent, not random!) ----

if ~isfield(options, 'preTrace')

preTrace = zeros(size(x));

postTrace = zeros(size(a1));

end

preTrace = (1 - tau_trace)*preTrace + tau_trace*x;

postTrace = (1 - tau_trace)*postTrace + tau_trace*a1;

% ---- Covariance component ----

cov_W1 = (a1 - mean_a1)*(x - mean_x)';

% ---- STDP kernel based on trace difference ----

dt = postTrace - mean(preTrace);

F = A_PLUS*(dt > 0).*exp(-abs(dt)) ...

- A_minus*(dt < 0).*exp(-abs(dt));

% ---- Final hybrid update ----

W1 = W1 + opts.eta_wake*(cov_W1.* (1 + F)) ...

- opts.eta_wake*opts(lambda)*W1;
% W2 hybrid update (simpler + more stable)
cov_W2 = (y_pred - mean_y)*(a1 - mean_a1)';
W2 = W2 + opts.eta_wake*cov_W2 ...
- opts.eta_wake*opts(lambda*W2;
end
% --- Optional supervised wake-step on output (delta rule for sigmoid) ---
if isfield(options, 'supervisedWake') && options.supervisedWake
% y_true for current sample must be scalar (0 or 1)
y_true = Y_target(:, idx); % 1 x 1
delta_out_wake = (y_true - y_pred).* (y_pred.* (1 - y_pred)); % scalar
dW2_wake = delta_out_wake * a1'; % 1 x n_hidden
% small backprop to hidden (optional, with smaller lr)
delta_h_wake = (W2' * delta_out_wake).* (1 - a1.^2); % n_hidden x 1
dW1_wake = delta_h_wake * x'; % n_hidden x nalgaps
% apply with separate small learning rates
if isfield(options, 'eta', 'supervisedWake') && ~isempty(options.etaent)
eta_sw = options.etaent;
else
eta_sw = 1e-4; % default small supervised learning rate
end
W2 = W2 + eta_sw * dW2_wake - eta_sw * options lambda * W2;
W1 = W1 + (eta_sw/5) * dW1_wake - (eta_sw/5) * options lambda * W1;
end
% --- Sleep phase ---
replay = rando sample(n_patterns, min(options.batch_size, n_patterns));
Xr = X(:,replay); Yr = Y_target(:,replay);
% --- Replayed activations and supervised consolidation (sigmoid + backprop) ---
A1r = tanh(W1 * Xr + b1) + 0.01 * randn(n_hidden, size(Xr,2)); % n_hidden x
batch
Yraw_r = W2 * A1r + b2; % 1 x batch (pre-activation)
Ypred_r = sigmoid(Yraw_r); % 1 x batch (post-activation)
E = Yr - Ypred_r; % 1 x batch
% output-layer delta (for sigmoid + MSE): delta_out = E.*sigma'(yraw)
delta_out = E.* (Ypred_r.* (1 - Ypred_r)); % 1 x batch
dW2 = (delta_out * A1r') / size(A1r,2);
% backprop to hidden: (n_hidden x 1) for each example -> (n_hidden x batch)
delta_hidden = (W2' * delta_out).* (1 - A1r.^2); % n_hidden x batch
dW1 = (delta_hidden * Xr') / size(A1r,2);
% Apply consolidation updates (use eta_sleep)
W2 = W2 + options.eta_sleep * dW2 - options.eta_sleep * options lambda * W2;
W1 = W1 + opts.eta_sleep * dW1 - opts.eta_sleep * opts lambda * W1;

% --- Monitor ---

preds = W2*tanh(W1*X + b1) + b2;

history(loss(epoch) = mean((preds(:)-Y_target(:)).^2);

history.energy(epoch) = norm(W1,'fro')^2 + norm(W2,'fro')^2;

if opts.verbose && (mod(epoch,10)==0 || epoch==opts.n_epochs)

fprintf('Epoch %3d/%3d | Loss = %.4f | Energy = %.4f\n', ...

epoch, opts.n_epochs, history(loss(epoch), history.energy(epoch));

end

end

%% === FIX D: Post-training evaluation (sigmoid-based) ===

sigmoid =@(z) 1/(1 + exp(-z));

Y_pred_cont = sigmoid(W2 * tanh(W1 * X + b1) + b2); % continuous prob

Y_pred_class = double(Y_pred_cont >= 0.5); % binary labels

conf Mat = confusionmat(Y_target(:), Y_pred_class(:));

overallAcc = mean(Y_pred_class(:)==Y_target(:)) * 100;

class0Acc = sum(Y_pred_class(:)==0 & Y_target(:)==0)/nnz(Y_target(:)==0)*100;

class1Acc = sum(Y_pred_class(:)==1 & Y_target(:)==1)/nnz(Y_target(:)==1)*100;

fprintf('\nPost-Training Evaluation:\n');

fprintf('Overall Accuracy: %.2f%%\n', overallAcc);

fprintf('Class 0 Accuracy: %.2f%%\n', class0Acc);

fprintf('Class 1 Accuracy: %.2f%%\n', class1Acc);

fprintf('Confusion Matrix:\n');

disp(conf Mat);

history.confusionMatrix = conf Mat;

history accuracy = struct('overall',overallAcc,'class0',class0Acc,'class1',class1Acc);

%% --- Output ---

history.W1 = W1; history.W2 = W2;

history.b1 = b1; history.b2 = b2;

history.fina1Loss = history(loss(end);

history.fina1Energy = history.energy(end);

fprintf('\nTraining complete. Final loss = %.4f | Final energy = %.4f\n', ...

history.fina1Loss, history.fina1Energy);

%% --- Performance Evaluation ---

% Compute final predictions on the whole dataset
%% --- Performance Evaluation ---

sigmoid =@(z) 1 ./ (1 + exp(-z));

Y_pred_cont = sigmoid(W2 * tanh(W1 * X + b1) + b2); % use sigmoid activation

if isfield, 'useAdaptiveThresh') && opts.useAdaptiveThresh

thr = median(Y_pred_cont(:)); % adaptive threshold

else

thr = 0.5;

end

Y_pred_bin = double(Y_pred_cont > thr);

% Overall accuracy

overall_ACC = mean(Y_pred_bin == Y_target);

% Class-wise accuracy

class0_ACC = mean(Y_pred_bin(Y_target==0) == 0);

class1_ACC = mean(Y_pred_bin(Y_target==1) == 1);

% Confusion matrix

conf Mat = confusionmat(Y_target(:), Y_pred_bin(:));

% Print metrics if verbose

fprintf('nModel Performance Summary:
');

fprintf('Overall Accuracy: %.2f%%
', 100*overall_ACC);

fprintf('Class 0 Accuracy: %.2f%%
', 100*class0_ACC);

fprintf('Class 1 Accuracy: %.2f%%
', 100*class1_ACC);

fprintf('Confusion Matrix:
');

disp(conf Mat);

% Store in history

history. accuracy.overall = overall_ACC;

history. accuracy.class0 = class0_ACC;

history. accuracy.class1 = class1_ACC;

history.confusionMatrix = conf Mat;

end

**Figure 1:** MATLAB Code for both the main calling script and the called function which evaluates the performance of the six Hebbian-like plasticity models.

## Results and Discussion

Table X shows the summary of model loss, model energy, and its overall accuracy percentage. The hybrid CovSTDP Hebbian model showed the maximum overall accuracy percentage.
<table><caption>**Table 3:** Summary of Model Performance.</caption><thead><tr><td>Rule</td><td>Loss</td><td>Energy</td><td>Overall Accuracy (%)</td></tr></thead><tbody><tr><td>Hebbian</td><td>0.5225</td><td>2.8464</td><td>73.75</td></tr><tr><td>Covariance</td><td>0.5176</td><td>2.6405</td><td>84.25</td></tr><tr><td>BCM</td><td>0.5471</td><td>1.1785</td><td>71.25</td></tr><tr><td>Oja</td><td>0.5336</td><td>2.4911</td><td>73.25</td></tr><tr><td>STDP</td><td>0.4604</td><td>1.9718</td><td>72.25</td></tr><tr><td>CovSTDP</td><td>0.4940</td><td>3.1080</td><td>85.25</td></tr></tbody></table>

Figure 2 shows the bar chart for the same results presented in Table 3.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** The bar chart for the six Hebbian-like plasticity models.</figcaption></figure>

The hybrid CovSTDP model performed the best, which is exactly what we would hope from a well-designed covariance + timing learning rule:

* CovSTDP achieved the highest accuracy (85.25%), surpassing all classic Hebbian-type rules.

* The Covariance rule is a close second (84.25%), showing that correlation-based learning is very suitable for this type of data.

* BCM and Oja are slightly weaker, which is expected because they are better at regulating stability than extracting temporal structure.
* STDP alone performs moderately, which is normal when the input is not fully spike-timing structured.

## Why the results are scientifically sound

We introduced temporal structure in the inputs:

$$ temporal\_mod = 0.3 * \sin(t); $$

$$ X = X + temporal\_mod; $$

This made the dataset time-correlated rather than purely random, which strongly favors models like CovSTDP that exploit both:

* Amplitude correlations (covariance)

* Temporal structure (STDP-like timing kernels)

This is exactly why the hybrid model outperforms the rest, which is excellent behavior from a modeling and neuroscience perspective.

The hybrid CovSTDP learning rule achieved the highest classification accuracy among all tested plasticity models. This suggests that combining covariance-based correlation learning with temporal spike-timing modulation provides a more powerful synaptic adaptation mechanism than either component alone. The improved performance is particularly evident when the input statistics contain structured temporal correlations, highlighting the advantage of integrating spatial and temporal learning signals in biologically inspired neural systems.

## Hebbian learning

The simple Hebbian rule showed moderate performance because it strengthens synapses based purely on raw co-activity between pre- and post-synaptic units. While this mechanism effectively captures basic correlations in the data, it lacks any normalization or temporal sensitivity, making it less capable of exploiting structured time-dependent patterns. As a result, Hebbian learning performs reliably but not optimally when temporal structure is embedded in the inputs.

## Covariance Hebbian rule

The covariance-based Hebbian rule performed strongly because it removes mean activity before weight updates, allowing the network to focus on statistically meaningful correlations rather than raw signal magnitude. This makes it particularly well-suited for datasets with structured variability, such as the temporally modulated Gaussian inputs.
However, without explicit timing sensitivity, it cannot fully exploit the temporal dependencies present in the data.

## BCM (Bienenstock-Cooper-Munro) rule

The BCM rule demonstrated moderate performance because its adaptive sliding threshold promotes stability and prevents runaway synaptic growth. This homeostatic mechanism improves robustness and generalization compared with simple Hebbian learning. However, BCM primarily regulates firing rates rather than explicitly modeling temporal correlations, limiting its ability to fully benefit from time-structured inputs.

## Oja's rule

Oja's rule showed moderate accuracy due to its built-in normalization term, which constrains weight growth and enables extraction of dominant principal components of the input. This makes the rule effective for variance-driven structure but less sensitive to temporal dynamics. Consequently, Oja's rule captures dominant spatial patterns in the data but does not fully leverage the temporal modulation embedded in the signal.

## STDP (Spike-Timing-Dependent Plasticity)

The STDP model achieved moderate performance because it relies primarily on spike timing differences to drive synaptic updates. The inputs were temporally modulated but not explicitly spike-coded, which limits the effectiveness of purely timing-based learning. As a result, STDP can partially exploit temporal structure but remains less effective than hybrid mechanisms that also incorporate statistical covariance information.

However, for the following type of input data, as shown in Figure 2, all six Hebbian plasticity models practically behaved the same as shown in Figure 3.

The six Hebbian models are trained on a synthetic dataset consisting of two simple, linearly separable Gaussian clusters in a 2-D feature space. Each cluster is generated from an isotropic normal distribution centered at distinct means, corresponding to class 0 and class 1. After normalization, a mild sinusoidal temporal modulation is added uniformly across samples to introduce a small degree of nonstationary variation. This produces clean but slightly perturbed clusters, ideal for demonstrating how different plasticity rules learn stable representations under simple yet biologically relevant input conditions.

### %% --- Built-in Demo Input Data (Gaussian Clusters + Mild Temporal Modulation)

n_per_class = 200;

% Gaussian cluster means

mu1 = [-1; 0.2];

mu2 = [1; -0.3];

% Cluster variances (isotropic)
sigma1 = 0.9;

sigma2 = 1.2;

% Generate the two Gaussian clusters

X1 = mvnrnd(mu1, sigma1 * eye(2), n_per_class);

X2 = mvnrnd(mu2, sigma2 * eye(2), n_per_class);

% Combine and normalize

X = [X1, X2];

X = X ./ max(abs(X), [], 'all');

% Binary class labels

Y_target = [zeros(1, n_per_class), ones(1, n_per_class)];

% Mild temporal sinusoidal modulation

t = linspace(0, 4*pi, size(X,2));

temporal_mod = 0.3 * sin(t);

X = X + temporal_mod;

<figure><img src="image_3.png" /><figcaption>**Figure 3:** Overall accuracy (%) of the six models trained on two linearly separable 2-D Gaussian clusters (classes 0 and 1), with a mild sinusoidal modulation added across samples.</figcaption></figure>

## Why All Six Plasticity Models Performed Similarly Well

All six plasticity rules, Hebbian, Covariance, BCM, Oja, STDP, and CovSTDP, converged to similar performance primarily because the input data are simple, linearly
separable Gaussian clusters mildly perturbed by a low-amplitude temporal modulation. The two synthetic clusters occupy well-separated regions in a two-dimensional space, and the hidden layer, with 15–40 tanh units, provides a highly overcomplete representation for such a low-complexity classification task. Under these conditions, all six rules generate weight updates that ultimately project the data onto a direction aligned with the class difference vector. Even though each rule modifies weights according to different biological principles, their effective gradient directions on this dataset are nearly identical, because the dominant statistical structure of the inputs is extremely strong and easy to extract.

A second reason for the similarity is that the wake–sleep architecture provides a powerful supervised correction mechanism during sleep, which partially “overwrites” differences between plasticity rules. During the sleep phase, mini-batch replay combined with sigmoid-based error backpropagation consolidates a stable decision boundary regardless of the specific unsupervised rule used during wake. This consolidation step acts as a unifying force: Even if Hebbian, BCM, Oja, or STDP drift divergently during online learning, the supervised replay repeatedly pulls the weights back toward the same solution manifold. As a result, all six plasticity mechanisms end up discovering essentially the same low-dimensional discriminant structure and therefore exhibit nearly identical accuracy, loss trajectories, and final energy levels on this dataset.

## Conclusions

Across all experiments using the simple, linearly separable Gaussian dataset with mild temporal modulation, the six plasticity mechanisms—Hebbian, Covariance, BCM, Oja, STDP, and CovSTDP—demonstrated remarkably similar overall performance. Despite their theoretical differences and biological interpretations, each rule successfully extracted the dominant statistical structure of the input: two well-separated clusters with minimal overlap. This indicates that, under clean and low-complexity input conditions, these learning mechanisms converge toward nearly identical synaptic configurations and classification boundaries. In other words, when the task is straightforward and the data geometry is simple, the choice of plasticity rule has little impact on final accuracy, loss reduction, or overall energy growth.

The key takeaway is that task complexity amplifies differences between learning rules, while simple tasks tend to mask them. In the present setting, all rules had more than enough capacity to represent the decision boundary, and the wake–sleep consolidation step further helped stabilize learning across plasticity types. Therefore, the results highlight an important principle for computational neuroscience and machine learning: *the discriminative power of a plasticity model only becomes apparent when the input statistics are rich, noisy, high-dimensional, or contain competing correlations.* When the
data remain simple and highly structured, even biologically diverse learning rules
collapse to a common solution, underscoring the robustness and redundancy inherent in
many forms of synaptic adaptation.

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