Towards Biologically Plausible Computing:
A Comprehensive Comparison

Hebbian learning is a fundamental concept in neural network models aimed at explaining how neurons in the brain adapt and learn from experience [12]. Proposed by [23], the theory suggests that the efficiency of a neuron in contributing to the firing of another neuron can increase if the two neurons are repeatedly involved in each other’s activation. The theory is often summarized as “Cells that fire together wire together.” It is biologically plausible and ecologically valid, relying solely on inputs entering the system to produce patterns of activity, without explicit tasks or teaching signals[24].

In neural network models, Hebbian learning is implemented through changes in the strength of connection weights between units. The strength of a connection weight determines the efficacy of a sending unit in activating a receiving unit. Through Hebbian learning, weights change as a function of the activity levels of the units involved. The basic form of the Hebbian learning rule can be expressed as:

where $\Delta w$ represents the weight change, $y(x,w)$, a function of the input and the weights, is the post-synaptic activation of the neuron $\eta$, determines the speed at which the weights change in response to unit activation. The activation level of a unit typically falls within the range of 0 to 1 and is calculated as a nonlinear function of the activation of other units and the strength of their connections to the unit.

The main problem of rule 1.1 is that it only allows weights to grow, not decrease. To prevent the weight vector from growing unbounded, it is possible to normalize it after every update, introducing a weight decay term proportional to the value of the weight, given by:

Equation (2) has the physical interpretation that at each iteration, the weight vector is modified by taking a step towards the input, with the size of the step being proportional to the similarity between the input and the weight vector. Consequently, if a similar input is presented again in the future, the neuron will be more likely to produce a stronger response. If an input (or a cluster of similar inputs) is repeatedly presented to the neuron, the weight vector tends to converge towards it, eventually acting as a matching filter. In other words, the input is memorized in the synaptic weights. From this perspective, the neuron can be seen as an entity that, when stimulated with a frequent pattern, learns to recognize it.

The abstract nature of the Hebbian learning rule allows it to be applicable to different types of neural networks and learning tasks. There can be multiple methods to implement Hebbian learning. HebbNet[25] is a neural network that utilizes an improved Hebbian approach with updated activation thresholds and gradient sparsity. SoftHebb[26] is a variant that combines standard deep learning elements with a Hebbian-like plasticity. It maintains a Bayesian generative model of the data without supervision and minimizes cross-entropy, offering advantages beyond traditional Hebbian efficiency.

In biological neural networks, signals are transmitted between neurons through spikes. The pre-synaptic neuron, responsible for signaling, generates a spike, releasing neurotransmitters within the synapse—a structure connecting it to the post-synaptic neuron, which, in turn, triggers the rise of the postsynaptic potential. As the potential accumulates and surpasses a certain threshold, a spike occurs in the postsynaptic neuron. The efficacy of pre-synaptic excitement in inducing post-synaptic excitement serves as the synaptic weight between two layers of neurons.

When a post-synaptic spike closely follows a pre-synaptic spike, it implies a strong association between the excitations of the two neurons, suggesting a causal relationship. In such cases, it is recommended to strengthen the synaptic weight between them. The shorter the interval between the spikes, the stronger the inferred causal relationship, leading to a greater increase in weight. Conversely, if the post-synaptic spike precedes the pre-synaptic spike, the intuitive assumption is that there is no causal relationship, resulting in a reduction in the synaptic weight. Spike-time-dependent plasticity (STDP), a learning rule within Hebbian learning models, adjusts synaptic weights based on the timing of spikes between pre- and post-synaptic neurons at the synapse, aligning with the aforementioned approach.

As proposed by [13], pre-synaptic excitation preceding post-synaptic excitation leads to long-term potentiation (LTP), resulting in an increased synaptic weight. Conversely, when pre-synaptic excitation follows post-synaptic excitation, it induces long-term depression (LTD), causing a decrease in synaptic weight. The magnitude of weight change is contingent upon the timing difference between spikes. This phenomenon, initially observed in neuroscience experiments, has been formulated as a learning rule for neuronal synapses. The mathematical expression for this rule is:

Where the $\Delta t=t_{post}-t_{pre}$ represents the time difference of spikes between the two neurons at the synapse. $\alpha_{+}$ and $\alpha_{-}$ determine the maximum amounts of synaptic modification and $\tau_{+}$ and $\tau_{-}$ are the time constants.

Spike-timing-dependent plasticity (STDP) manifests across a variety of species. Following an initial discussion of STDP, we transition to an exploration of its associated biological mechanisms, particularly the $Ca^{2+}$ hypothesis[27, 28]. Notably, a high influx of $Ca^{2+}$ in the post-synaptic neuron induces Long-Term Potentiation (LTP), while a moderate influx results in Long-Term Depression (LTD). Distinct $Ca^{2+}$ levels activate specific molecular pathways; high $Ca^{2+}$ activates CaMKII for LTP, while moderate levels activate PP1 and calcineurin for LTD[29]. The level of $Ca^{2+}$ influx hinges on the interval between the excitatory postsynaptic potential (EPSP) triggered by the pre-synaptic neuron’s activation and the backpropagating action potential (BAP) initiated by the post-synaptic neuron’s activation. Positive intervals signify BAP occurring shortly after EPSP, while negative intervals denote BAP preceding EPSP. When the interval is positive, the BAP leads to $Mg^{2+}$ unblocking NMDA receptors[30]. Simultaneously, EPSP deactivates specific ion channels, augmenting BAP magnitude[31, 32]. This intensified BAP activates voltage-dependent $Ca^{2+}$ channels (VDCCs)[33]. The interplay between EPSE and BAP in positive intervals results in a substantial $Ca^{2+}$ influx in the post-synaptic neuron. Conversely, when the interval is negative, the afterdepolarization of BAP coinciding with EPSP induces a moderate $Ca^{2+}$ influx[29]. BAP, causing $Ca^{2+}$ influx through VDCCs, inhibits NMDA receptors[34]. These interactions between BAP and EPSP in negative intervals lead to a moderate level of $Ca^{2+}$.

The STDP learning rule has many different mathematical forms and boasts numerous variations. Rather than computing $\Delta w$ between the current spike and every single spike in the past when a spike occurs at the synapse, a more efficient approach involves maintaining a synapse trace for both pre- and post-synaptic neurons. These traces undergo continuous modification over successive timesteps:

Where $t_{i-1}$ and $t_{i}$ are two adjacent timesteps. $x$ stands for the synapse trace and decays over time by a coefficient $\lambda$ ($0<\lambda<1$). $s^{(t_{i})}=1$ when a spike occurs at $t_{i}$, and $s^{(t_{i})}=0$ if no spike occurs.

Numerous studies on the effects of Spike-Timing-Dependent Plasticity (STDP) through virtual neural network simulations have been conducted. Research indicates that STDP confers selectivity to neurons towards different input signals [3] and fosters synchronous neuron activities [35][36]. Notably, the weights between pre- and post-synaptic neurons, influenced by STDP, typically exhibit a bimodal distribution. Moreover, STDP demonstrates the ability to maximize mutual information between input and output neuron spikes [37]. Applying the STDP learning rule enables a single neuron to detect a single hidden repeating spatiotemporal pattern [38], while multiple neurons can selectively detect different patterns [39] from a large set of spike trains with inherent noise. This suggests that STDP serves as an unsupervised learning algorithm with the capability to handle time-series data.

Significant efforts have been devoted to applying the STDP algorithm, primarily through spiking neural networks, to various machine learning tasks, particularly in the field of image processing. Experimental datasets include well-known sets such as MNIST, CIFAR-10, Caltech face, and motorbike datasets. Diverse neural network architectures have been employed, ranging from hierarchical networks inspired by the human visual cortex [40][41][42], single-layer fully connected networks with inhibitory neurons [43][44], to deep convolutional networks [45] and liquid state machines [46]. It is noteworthy that certain techniques are commonly employed across these studies. For instance, preprocessing of the original image, including the use of neuromorphic datasets and the application of Difference of Gaussian filters, specific receptive field configurations, latency encoding, and lateral inhibition, have been integral components in many works.

In practical experiments, we usually use equation3 to apply STDP, but this learning rule only be found in a small wide brain area in specific environments. There are two classes of neurons which can be called excitatory and inhibitory neurons playing an important role in the processes of learning, the STDP between them are apparently different, additionally, STDP between the neurons from the same position of the brain in different species also be different[47]. the trigger patterns of STDP in vivo are also different from the binary pair pattern from equation3[48], such as the different frequency burst to apply in pre-synaptic neuron, the triple pairs of spikes between pre-synaptic neuron and post-synaptic neuron and the pairs of burst and spike. The STDP in vivo also be influenced by the dendritic position which the post-synaptic membrane is in[49]. In addition to the spiking of the pre-and postsynaptic neurons, STDP is also regulated by other inputs. In particular, neuromodulators and inhibitory activity in the network can affect both the magnitude and the temporal window of STDP[50]. So, these phenomenons in vivo prove that the STDP we use might not fit in the biological plausibility.

The concept of Feedback Alignment (FA), originally proposed by Lillicrap et al. [14], presents an intriguing paradigm in neural network training. One of FA’s remarkable attributes lies in its resolution of the weight transport problem, as elucidated in [51], aligning it more closely to the mechanisms observed in the human brain. Subsequent research derived from FA has not only facilitated the localization of error signal transmission [52], but also delved into aspects like update locking [53], further enhancing its biological plausibility.

Diverging from conventional backpropagation methods, FA shows that one can achieve similar effects and accuracy on classification tasks by replacing feedback weights with fixed random synaptic weights. Central to this theory is the notion that the precise alignment of feedback weights to the transpose of the forward weights $W^{T}$ is unnecessary during training. Instead, a designated weight matrix $B$ suffices in steering the network in roughly the same direction as backpropagation, thereby facilitating network training. The replacement matrix $B$ chosen only has to ensure the fulfillment of the following equation on an average basis:

Where $e$ denotes the error of the network’s output, and $W$ represents the synaptic weights of the forward path. However, studies [54] have revealed a notable decline in FA’s performance when employing deeper convolutional architectures, in contrast to its comparable performance to BP in simpler MLP networks.

Direct Feedback Alignment (DFA) [52] , an offshoot of FA, emerges as a prominent avenue of research aimed at addressing this limitation. DFA tackles this challenge by introducing direct feedback paths and distinct fixed random weights for each hidden layer. This strategic design localizes learning signals by establishing direct connections between errors and individual layers, bypassing the conventional layer-by-layer backpropagation from the output. Consequently, training deeper networks becomes more feasible, attributed to the disentangled feedback paths offering increased flexibility in transmitting error signals. Learning is thus regarded as an extension of a forward pass, marking DFA as a notable stride towards biological plausibility.

To formalize it, traditional FA updates previous hidden layers based on the change of the following layer, depicted as:

While DFA directly updates each hidden layer in accordance to distinct weight matrices and the error of the output layer:

Here $B_{i}$ represents a fixed random weight matrix, $\odot$ indicates element-wise multiplication, and $f^{\prime}$ denotes the derivative of the non-linearity function of hidden layers.

Empirical evidence underscores the efficacy of DFA, notably in reducing training time and narrowing the accuracy gap compared to BP on datasets like MNIST and CIFAR [53]. Owing to its simplicity and effectiveness, DFA is widely adopted as a foundational model in subsequent research endeavors. Nevertheless, experimental outcomes [55] remain somewhat limited when scaling to more intricate network architectures or larger datasets like ImageNet.

The backpropagation algorithm lacks biological plausibility, as in the human brain, biological neurons are interspersed with linear and nonlinear elements[56]. Utilizing backpropagation through feedback paths for the propagation of credit assignment necessitates precise knowledge of the nonlinear derivatives employed in the corresponding feedforward computations. Additionally, it requires alternating between exact feedforward propagation and backpropagation processes across different neuronal layers. This mechanism of gradient communication and weight transfer is biologically impractical[57].

Inspired by earlier research, Bengio et al. [58] proposed Target Propagation (TP) as a novel credit assignment approach in response to the challenges posed by the backpropagation method. TP assigns a target value $\hat{h_{l}}$ to each layer $l$, rather than employing a loss gradient. These target values are designed to be in close proximity to the activation values, with the potential to yield a reduced loss if achieved during the feedforward phase.

A distinct aspect of Target Propagation[59] is that its backward pass operates within the same dimensional framework as the forward-pass neural activities. The objective in this phase is to align the layer activities with those induced backward, thereby facilitating the generation of the desired output. Upon receiving an input, the final output layer undergoes feedforward propagation and is directly optimized to minimize the loss. In contrast, the remaining layers are oriented towards aligning with their assigned target values. The training process involves two types of losses at each layer level. Inverse loss is used to train an approximate inverse, which is parameterized in a manner analogous to the forward computation:

where $g$ is the approximate inverse: $g(h_{l};\lambda_{l})=\sigma_{l}(V_{l}h_{l}+c_{l}),\lambda_{l}=\{V_{l},c_{l}\}$. Forward loss imposes a penalty on layer parameters that result in activations divergent from their designated targets:

These losses are localized, impacting only the parameters of the individual layer, and do not take into account any implicit dependencies on the parameters of other layers.

Vanilla Target Propagation calculates targets by back-propagating the targets from higher layers through layer-specific inverses:

However, this simplistic approach may encounter difficulties in scenarios where different instances of the same class present diverse appearances. In such cases, TP tends to enforce uniformity in their representations across all layers, including the early ones. To address this issue, the Difference Target Propagation (DTP) was introduced, incorporating linear correction terms into the feedback process:

The second term is the reconstruction error, providing a linear stabilizer for the inaccuracies in inverse functions. This enhancement significantly improves the recognition performance of the Target Propagation method. In the seminal study by Lee et al. [59], the target for the penultimate layer was determined using network loss gradients, deviating from standard TP methods. [60] propose the Simplified DTP (SDTP) as a refinement to DTP, where the target for the penultimate layer is computed according to eq.10. This modification effectively eliminates the biologically unrealistic aspects of gradient communication and associated weight-transport in the TP algorithm.

Originally proposed by [16], predictive coding (PC) is an influential theory in computational and cognitive neuroscience. The central idea of the theory is that the brain is composed of a hierarchy of layers, while high layers predict the activities of adjacent low layers. The entire brain maintains a cognitive model of the world, activities that cannot be accurately predicted are regarded as prediction errors which will be transmitted upwards for high-level to process. Over time, the synaptic connections between high and low levels are updated until the prediction error of the entire system is minimized. While predictive coding originated in theoretical neuroscience as a model of information processing in the cortex, recent work has developed the idea into a general-purpose algorithm able to train neural networks using only local computations. Compared to the classical backpropagation algorithm, hierarchical predictive coding models are considered to be more biologically plausible. When Rao and Ballard first proposed the hierarchical predictive coding model in 1999, they cautiously suggested the biological interpretability of predictive coding by indirectly highlighting similarities between the hierarchical predictive coding model and classical effects such as endstopping in the visual processing regions of the brain cortex [16]. Since then, the emergence of predictive coding theory has, to some extent, challenged cognitive neuroscience. Consequently, a substantial amount of neuroanatomical research has been conducted to validate or refute predictive coding. This research includes a wealth of anatomical and physiological evidence supporting predictive coding as an assumption for information transmission within the cortical hierarchy, especially in early visual processing [61]. However, despite abundant indirect evidence suggesting the potential existence of predictive coding mechanisms in the cortical regions of the brain, crucial direct evidence is still lacking. Caution should be exercised in assessing the biological plausibility of predictive coding.

Hierarchical predictive coding networks (PCNs) composed of $L$ hidden layers include two kinds of neurons. $x_{l,t}$ denotes the neuron that encodes time-depend predictions in layer $l$ at time $t$, $\epsilon_{t}$ denotes the neuron that compute prediction errors in layer $l$ at time $t$($l\in\{0,...L-1\}$). In such cases, the input signal is transmitted from low-level to high-level, and high-level neurons predict the value from the following layer according to:

where $f$ is a nonlinear function, and $\theta_{l+1}$ denotes the matrix of weights connecting layer $l+1$ to layer $l$. Prediction error represents the difference between actual activity and its prediction, which is denoted by $\epsilon_{l,t}=x_{l,t}-\mu_{l,t}$. The errors are then propagated down the hierarchy and used in the learning process to update the weights of the network. Ultimately, the learning algorithm optimizes a global energy function, defined as the sum of squared prediction errors at each layer:

During training, the highest layer is fixed to an input data point, and the lowest layer is fixed to a label or target vector. During a process called inference, the weight parameters are fixed, and the neural activities are continuously updated to minimize the energy function by running gradient descent until convergence, at which point a single weight update is performed. During the weight update, the value nodes are fixed, and the weight parameters are updated via gradient descent on the same energy function. When defining inference and weight update this way, every computation only needs local information to be updated [62].

During testing, only the lowest layer is fixed to the data, so the network infers the label given a test point. This process is equivalent to the inference phase described above: the weight parameters are fixed, and the neural activities are updated until convergence by running gradient descent on the energy function. Note that different works follow different paradigms for the order of updating the neural activities $x_{t}$ and weights $\theta_{t}$. In most works, neural activities are all simultaneously updated for T iterations with the aim of reaching convergence of the inference process, and the weights are updated once upon convergence [63, 62, 64]. However, in certain cases, it has been noted that updating the weights and activities of different layers in different moments yields a better performance [65, 66].

PCNs can be mathematically derived as variational inference on hierarchical Gaussian generative models. The hierarchical model consists of multiple layers indexed by $l$.The distribution of activations at each layer can be approximated as Gaussian distribution with a mean given by a nonlinear function $f(x_{l+1})$ with parameters $\theta_{l+1}$ and an identity covariance $I$,

For example, consider a scenario where the input layer $l=0$ is held constant at a specific data item$s_{in}$. Our objective is to deduce the state of the remaining network based on this conditioning $p(x_{0},...,x{L-1}|x_{L})$. Solving this inference problem can be accomplished through variational inference. Broadly speaking, variational inference tackles the challenge of approximating an intractable inference by framing it as an optimization problem. This involves optimizing the parameters of an approximate variational posterior distribution $q$ to minimize its divergence from the optimal posterior $p$.The optimization process revolves around minimizing an upper bound on this divergence, referred to as the variational free energy $\mathcal{F}_{t}$. In PCNs, our assumption is that the variational posterior is factorized into independent posteriors for each layer $q(x_{0,t},...,x_{L-1,t})=\prod\limits_{l=0}^{L-1}q(x_{l,t})$. When combined with the Laplace approximation, this simplification enables us to express the free energy as a sum of squared prediction errors.

As we mentioned above, $\epsilon_{l,t}=x-\theta_{l+1}f(x_{l+1,t})$ is the prediction error for each layer.When applied to ANNs, we generally operate under the assumption that the dependencies between layers are characterized by a parameter matrix $\theta_{l+1}$. This matrix corresponds to the weights within an ANN. Consequently, updates to both the activations $x_{l,t}$ and weights can be performed through gradient descent on the free energy.

The operation of PCNs involves two distinct phases. Initially, the activation $x_{l,t}$ undergoes updates to minimize the free energy until they attain equilibrium. Subsequently, the weights $\theta_{l}$ undergo a single-step update based on the equilibrium values of the activations $x_{l,t}^{*}$. These phases are commonly referred to as inference and learning.

PC converts the feedforward pass of an ANN (artificial neural network) into an inference problem, which requires manipulating the activations of the ANN layers under certain constraints on the input or output layers, or both. The uncertainty about the optimal activations is represented by a Gaussian distribution with a mean given by the top-down prediction from the higher layer. Importantly, this inference problem is solved dynamically in each inference stage. Furthermore, the conditioning variables can be modified adaptively according to the task demands. During execution, the PCN (predictive coding network) utilizes its learned generative model embedded in the weights $\theta_{t}$, enabling it to cope with diverse inference problems. This demonstrates the enhanced flexibility of PCNs over ANNs, as evidenced by recent studies.

A common assumption is that PC provides the best inversion method for hierarchical Gaussian generative models. However, this assumption does not hold when compared to the state-of-the-art generative AI systems that employ deep neural networks, which can achieve superior performance in various domains. Moreover, there are several aspects of cortical microcircuitry remain which have so far resisted simple interpretation while using the PC framework. As such, predictive coding might be ’right’ in some sense, but still missing core aspects of the computation that actually goes on in the cortex. Memory is another vital function of the brain. It seems that cortical areas implement short-term and long-term memory by means of persistent neural activity. However, modeling these memory processes within a predictive coding framework poses significant challenges.

Proposed by Geoffrey Hinton, the Forward-Forward algorithm is a new learning procedure to update the weights of the network which is more biologically plausible than backpropagation. Compared with the backpropagation algorithm, the Forward-Forward algorithm updates the weights of each layer with two forward propagation processes, specifically, it updates network weights layer by layer in place, without the need to store neural activities or wait for the backpropagation of error gradients. In addition, it does not require full knowledge of the forward calculation function therefore it can tolerate black-box functions. These properties show greater biological plausibility, more akin to the functioning of the cerebral cortex. So this algorithm deserves our attention.

With the advantages of the algorithm established, it’s pertinent to explore its basic design. Generally, during the training phase, the algorithm will use two forward passes to replace the forward and backward passes, one with positive data and the other with negative data. Each layer has its own objective function, as a result, the train of each layer is independent, and the parameter updates of the former layers do not depend on the activities of the neurons in the layers behind them. The training target of each layer is to increase the goodness of positive data and decrease the goodness of negative data. In other words, the training objective for each layer is to be able to differentiate between positive data and negative data. For a given layer, The probability that an input vector is positive can be formulated as below:

where $y_{j}$ is the activity of hidden unit $j$,$\theta$ is a threshold, and $\sigma$ is the logistic function.

Nowadays the Forward-Forward algorithm shows comparable speed and accuracy on some small problems[17], but for large models, its performance is inferior to the backpropagation algorithm. Additionally, the Forward-Forward algorithm still has points that do not satisfy biological plausibility, for example, the output of each layer is still continuous signals rather than spikes, and the error signals are unsigned. In conclusion, while the Forward-Forward algorithm exhibits significant strengths and an innovative design in terms of biological plausibility, it is not without limitations. This, however, does not diminish its potential. It is the hope that this exploration of the Forward-Forward algorithm will contribute to further understanding of biological plausibility.

The concept of “perturbation learning” was first proposed by [18], which presented an associative reinforcement learning algorithm for networks containing stochastic units. Then, [19] introduced a perturbation learning paradigm to linear feedback neural networks. In recent years, perturbation learning methods for training neural networks mostly mean utilizing a random perturbation on network components. If we can observe a decrease in the error, this perturbation will be accepted. Otherwise, we will reject this perturbation and try another perturbation. There are generally two perturbation learning methods nowadays: weight perturbation and node perturbation. While perturbation learning may necessitate multiple iterations for neural networks to achieve convergence, it can be considered a biologically plausible training methodology. This characterization arises from the absence of direct gradient descents guided by global targets in the process.

The local Losses method [71, 15] represents a paradigm shift from global error correction to layer-specific training. Instead of propagating a single global error signal backward through the entire network, Local Losses employs local error signals for each layer. Each layer is equipped with its own classification layer that generates an error signal used to update that specific layer’s parameters. This approach aligns more closely with biological neural processes, where learning appears to be more distributed and locally governed.

Formally, consider a neural network composed of $L$ layers. Let $h^{l}$ denote the activations of the $l$-th layer, where $h^{l}=f^{l}(h^{l-1})$ and $f^{l}$ represents the transformation applied by layer $l$. In traditional back-propagation, the overall loss $\mathcal{L}_{\text{global}}$ depends on the final output $h^{L}$ and the target labels $y$:

Gradients of this global loss with respect to each layer’s parameters are computed and propagated backward from the output layer to the input layer. Conversely, in the Local Losses method, each layer $l$ has its own local loss function $\mathcal{L}^{l}$, which depends on the activations $h^{l}$ and a set of auxiliary target labels $y^{l}$:

The auxiliary targets $y^{l}$ can be derived from the true labels $y$ or be unsupervised signals appropriate for the task at hand.

Each layer $l$ is paired with a classifier $C^{l}$, which maps the activations $h^{l}$ to a prediction $\hat{y}^{l}$:

The local loss $\mathcal{L}^{l}$ for layer $l$ can be written as:

The parameters of both the transformation $f^{l}$ and the classifier $C^{l}$ are updated based on the gradient of this local loss:

where $\theta^{l}$ represents the parameters of layer $l$, and $\eta$ is the learning rate.

This layer-specific training allows each layer to learn independently, guided by its own local loss. This method mimics the potentially distributed nature of learning in the brain, where different regions may adapt based on localized feedback. The Local Losses approach offers several benefits, including biological plausibility, scalability, and robustness. However, this approach also presents challenges: 1. Auxiliary Target Design: Designing appropriate auxiliary targets $y^{l}$ for each layer can be complex and may require domain-specific knowledge; 2. Coordination: Ensuring that independently trained layers work harmoniously to achieve the overall task requires careful architectural and procedural design.

The continuous Hopfield models [21] are recurrent neural networks that serve as content-addressable memory systems. An appropriate “energy” function is constructed that is always decreased by any state change produced by the activity of each neuron. The energy function’s minima correspond to the preferred states of the model. The examination of iterations of Hopfield models continues to be a subject of ongoing research over several decades, including the exploration of diverse learning mechanisms. Hebbian learning and the Storkey learning rule represent two established traditional learning approaches in Hopfield models, while in recent years, a novel learning paradigm known as equilibrium propagation [22] has been introduced.

The combination of Equilibrium Propagation with Hopfield models is regarded as having significant biological plausibility. Hopfield models are commonly seen as approximations of human memory systems, deviating from the typical layered architecture observed in most artificial neural networks (ANNs). The weight update functions of Equilibrium Propagation can be constructed by synaptic learning rules based on pre- and post-synaptic activities. While this algorithm, akin to Back-propagation, calculates the gradient of the objective function during the second phase, it is notable that the neural computations in both phases are consistent, enhancing the possibility of a biological implementation.

Equilibrium Propagation comprises a two-phase training process. In the initial phase, a prediction is generated by fixing the inputs and allowing the Hopfield network to converge to a local energy function minimum. Subsequently, during the second phase, the outputs are adjusted toward their desired targets, and the network converges to a new state with a marginally reduced prediction error. Temporal derivatives of the neural activities in Equilibrium Propagation and Back-propagation have been proved equal by [72]. This algorithm also exhibits a connection to contrastive Hebbian learning, as it learns the second phase fixed point by reducing the total energy and in comparison to the first phase fixed point achieved through prediction. In the wake of the introduction of Equilibrium Propagation, it has been extended to encompass Convolutional and Spiking Neural Networks [73, 74], highlighting a compelling approach as a biologically plausible method for gradient computation in deep neural networks. However, an existing issue is that Hopfield models require symmetric connections between nodes, and there is yet no biological evidence supporting the existence of a symmetric connection among actual neurons in the brain [22].

In this section, we will introduce the datasets we utilized

MNIST  The MNIST dataset consists of 70,000 grayscale images of handwritten digits, divided into 60,000 training images and 10,000 testing images. Each image is 28x28 pixels, and the digits range from 0 to 9. This dataset is a benchmark in the field of machine learning and image classification due to its simplicity and ease of use, providing a solid baseline for evaluating the performance of different algorithms.

CIFAR  The CIFAR dataset, created by the Canadian Institute For Advanced Research, is a widely-used benchmark in machine learning and computer vision, consisting of two versions: CIFAR-10 and CIFAR-100. CIFAR-10 contains 60,000 32x32 color images in 10 classes, split into 50,000 training and 10,000 testing images, covering categories such as airplanes, cars, and animals. CIFAR-100 extends this to 100 classes grouped into 20 superclasses, with the same train-test split ratio.

Haxby  The Haxby dataset is a neuroimaging dataset that records brain activity using functional Magnetic Resonance Imaging (fMRI) while subjects view images from different categories. This dataset is particularly valuable for our research as it allows us to compare the feature representations learned by biologically plausible algorithms with actual brain activity patterns. The dataset includes multiple subjects and a variety of visual stimuli, making it ideal for studying how well machine-learning models can mimic human brain processing.

By utilizing these datasets, we aim to provide a comprehensive evaluation of the algorithms’ performance across simple and complex visual tasks, as well as their ability to replicate human brain activity patterns.

In our experiments, we employed two primary types of models: Convolutional Neural Networks (CNNs) and Multi-Layer Perceptrons (MLPs). The CNN model consists of three convolutional layers with kernel sizes of $5\times 5$, $5\times 5$, and $3\times 3$, respectively, and channel sizes increasing from input to 64, 128, and 256, with a final fully connected (FC) layer mapping to the output classes. Each convolutional layer is followed by a max pooling layer with a kernel size of $2\times 2$. The activation function used throughout the CNN is ReLU. The MLP model is composed of an input layer, two hidden layers with 1024 and 256 units respectively, and an output layer, all utilizing ReLU activation functions. Both models were trained using a standard backpropagation algorithm with categorical cross-entropy loss and optimized with stochastic gradient descent (SGD) with momentum. Hyperparameters, such as learning rate and batch size, were fine-tuned based on validation set performance. This implementation enabled us to evaluate and compare the performance of biologically plausible algorithms across different network architectures and datasets, providing insights into their effectiveness in mimicking human brain activity patterns.

Specifically, for the STDP method, we utilize the Pytorch-based framework called SpikingJelly [75] which is tailored for spiking neural networks.

We evaluate all bio-plausible algorithms on image classification benchmarks, reported in Table 2.

As shown in Table 2, we can draw conclusion that: (1) Local losses, being the most similar to backpropagation, achieve performance closest to traditional backpropagation. This aligns with our expectations, as layer-by-layer training of neural networks still relies on global label signals, utilizing local errors exclusively for weight updates. (2) While Hebbian learning and STDP are the most biologically plausible algorithms, they exhibit significantly poorer performance compared to alternative methods. Particularly, they both fail to converge on the CIFAR-100 dataset. (3) The forward-forward algorithm and equilibrium propagation (an energy-based method) appear effective primarily with the MLP architecture on tasks such as MNIST. However, these methods encounter challenges and fail to converge when applied to more complex tasks or architectures like CNNs.

Taking Table 1 into consideration, it is evident that the current bio-plausible learning algorithms, while showing promise, often fall short of achieving the high performance demonstrated by traditional backpropagation, especially on more complex datasets like CIFAR-10 and CIFAR-100. Algorithms like Feedback Alignment, Direct Feedback Alignment, Predictive Coding, and Local Losses demonstrate competitive performance on simpler datasets such as MNIST, indicating their potential viability. However, their diminished effectiveness on more challenging tasks underscores the need for further advancements in this field.

For the community to make significant strides, it is necessary to develop a learning algorithm that seamlessly integrates biological plausibility with high performance. Such an algorithm would need to address the key criteria of biological learning, including asymmetry of forward and backward weights, local error representation, non-parallel training, realistic neuron models, and unsigned error signals, without compromising on accuracy and scalability. This endeavor will likely involve innovative approaches that combine insights from neuroscience with advanced machine learning techniques.

The development of such an algorithm would not only bridge the gap between biological realism and computational efficiency but also potentially lead to more robust, efficient, and adaptable learning systems. It would pave the way for a new era of machine learning that is not only inspired by but also closely aligned with the principles of biological learning, ultimately enhancing our ability to develop intelligent systems that can learn and adapt in more human-like ways.

To investigate the biological plausibility of the previously discussed algorithms, we have developed a method to compute representation similarity between biologically plausible algorithms and the human brain. Specifically, we utilize NeuroRA, a toolbox designed for representation similarity analysis. NeuroRA allows us to acquire various types of neural data, such as fMRI and ROI, and to compute the Representational Dissimilarity Matrix (RDM). Using the Haxby dataset, which is compatible with NeuroRA, we train our model and compute an average representation for each category within the dataset. Based on these average representations, we calculate the RDM for our model. Subsequently, we measure the similarity between the RDMs of human brains and our model using cosine similarity, which serves as our similarity metric. The results are presented in Table 3.

Note that we chose $4$ algorithms because they were the only ones that successfully converged during training. For CNN trained by the forward-forward algorithm, it fails to converge. From Table 3, we find that: (1) Backpropagation significantly outperforms other bio-plausible algorithms on accuracy. This observation is reasonable and aligns with findings in datasets like MNIST and CIFAR. (2) For RDM similarity, Backpropagation is not the top performer; predictive coding shows the highest similarity. This suggests that although backpropagation achieves high accuracy, it might not be as biologically plausible in terms of the representation it learns. (3) There appears to be a trade-off between accuracy and similarity. Backpropagation achieves relatively high RDM similarity alongside its superior performance, suggesting that high accuracy might contribute to higher similarity. This leads us to speculate that if the performance of algorithms like predictive coding and forward-forward were improved to match the accuracy of backpropagation, they might exhibit even higher similarity scores. This hypothesis indicates that enhancing the accuracy of biologically plausible algorithms could potentially increase their alignment with human neural representations, offering a promising direction for future research in developing models that are both effective and biologically plausible.

Biologically inspired models often face challenges in scalability due to their intricate architectures and the need to adhere to the constraints of biological plausibility. Traditional deep learning models have demonstrated remarkable success in handling complex, high-dimensional tasks, such as image and speech recognition, but they often rely on computational techniques and resources that are not biologically feasible. To bridge this gap, researchers must explore methods to enhance the scalability of biologically plausible models without compromising their inherent principles. This involves investigating new algorithms, optimizing hardware implementations, and perhaps most critically, developing hybrid approaches that combine the strengths of both biologically inspired and traditional deep learning methods.

Biological networks excel at processing temporal sequences, a capability that remains a significant challenge for artificial systems. The dynamic and recurrent nature of biological neurons allows for sophisticated temporal processing, enabling organisms to navigate, predict, and learn from sequential events in their environments. In contrast, most artificial neural networks, particularly those used in deep learning, struggle with temporal dependencies unless specifically designed with recurrent or attention-based mechanisms. Developing biologically plausible methods for learning temporal sequences could involve leveraging the properties of spiking neural networks (SNNs) or other neuromorphic computing approaches that naturally accommodate time-dependent processing. Additionally, understanding how biological networks balance short-term and long-term memory and applying these principles to artificial systems could lead to more robust and efficient temporal learning models.

Biological systems are not only efficient in learning but also in their energy consumption and adaptive capabilities. The optimization of neural circuit dynamics to support efficient learning is a multifaceted problem that encompasses the minimization of energy use, the maximization of learning speed, and the enhancement of adaptability to changing environments. Biological neurons exhibit a variety of dynamic behaviors, such as synaptic plasticity and homeostasis, which contribute to their learning efficiency. Translating these dynamics into artificial systems requires a deep understanding of the underlying biological processes and the development of algorithms that can mimic these processes effectively. This might involve creating more sophisticated models of synaptic plasticity, incorporating adaptive mechanisms that allow artificial networks to self-tune in response to environmental changes, and designing hardware that can support these dynamic processes efficiently.

The development of fully biologically plausible algorithms is a crucial area of research within the field of biologically inspired deep learning. These algorithms aim to replicate the principles and mechanisms of learning observed in biological systems, such as local learning rules, Hebbian plasticity, and the balance between excitatory and inhibitory signals. Achieving full biological plausibility involves not only mimicking the structural and functional aspects of biological neurons but also adhering to the constraints of biological systems, such as limited energy resources and real-time processing capabilities.

The advancement of hardware specifically designed for biologically plausible models is vital for the practical application of these systems. Neuromorphic chips, such as those designed for spiking neural networks (SNNs) [76], represent a significant step towards this goal. These chips, like IBM’s TrueNorth [77] and Intel’s Loihi [78], aim to emulate the structure and functionality of biological neural circuits, enabling more efficient processing of information through event-based computation and asynchronous communication. Unlike traditional processors, neuromorphic hardware can inherently handle the temporal dynamics and adaptive behaviors characteristic of biological systems. This includes the ability to support synaptic plasticity, dynamic learning, and low-power operation. Developing and optimizing such hardware involves overcoming challenges related to scalability, integration with existing computing infrastructure, and ensuring that the hardware can support the complex dynamics required for advanced learning tasks. By focusing on these areas, researchers can create more effective and efficient hardware solutions that bring biologically plausible deep learning closer to practical reality.