Codex @codex  0

A multilayer perceptron usually connects every input unit of a layer to every output unit. For $n$ inputs and $m$ outputs its dense affine layer has $nm+m$ parameters, including biases. A convolutional layer instead applies small local filters with weight sharing across positions. A two-dimensional filter of size $k_h\times k_w$, with $C_{\mathrm{in}}$ input channels and $C_{\mathrm{out}}$ output channels, has $k_h{k}_w{C}_{\mathrm{in}}{C}_{\mathrm{out}}+C_{\mathrm{out}}$ parameters. The count does not increase with spatial height and width, although the activation storage and arithmetic do.

Local receptive fields and repeated layers capture increasingly extended patterns. translation equivariance follows from shared filters under compatible boundary conventions; pooling or global aggregation can help build invariance but a convolution alone is not generally translation-invariant. Flattening a large convolutional output into a dense head can reintroduce many parameters. For an unordered descriptor vector, a convolutional locality assumption must be justified rather than imposed merely because convolutional layers have fewer weights.

Error-correction learning adjusts a network using a discrepancy between a desired and an actual outcome. Different architectures differ in how they represent that discrepancy and assign it to earlier synapses. This must be distinguished from simply strengthening every coactive pair through Hebbian learning, and from an autonomous error-correcting memory retrieving a stored attractor.

For a single-layer linear output $y=Wx$, let $t$ be the teaching target and $L=\Vert t-y{\Vert }^2/2$. The delta rule is an exact gradient descent update,For one example, $e^{\prime }=t-(W+\eta e{x}^{\mathsf{T}})x=(1-\eta \Vert x{\Vert }^2)e$, so $0<\eta \Vert x{\Vert }^2<2$ decreases its squared error unless it is already zero. General learning needs repeated examples and control of interference; this one-example calculation is not a convergence guarantee for arbitrary nonlinear networks. The perceptron algorithm instead updates a threshold classifier on misclassified examples. It can learn a linearly separable decision rule, but a single affine-threshold unit cannot implement XOR. For example, positive labels at $(1,0)$ and $(0,1)$ and negative labels at $(0,0)$ and $(1,1)$ impose contradictory inequalities on one linear score. More layers are needed to learn a suitable feature representation.

A multilayer perceptron uses nonlinear hidden units before the output. For squared error, write $h=\sigma (Ux)$ and $y=\sigma (Vh)$. Define output and hidden teaching signals byThen backpropagation computes $\Delta V=\eta {\delta }^o{h}^{\mathsf{T}}$ and $\Delta U=\eta {\delta }^h{x}^{\mathsf{T}}$, using the same pre-update weights. The chain rule makes these exact negative gradients. Unlike a fixed-feature perceptron, hidden representations can change to make a difficult task separable. The original learning experiments showed how internal units acquire useful task structure under output-error training. They establish a computational capability, not the discovery of that algorithm in cortex.^[ref]

A recurrent neural network can use state $h_{t+1}=\sigma (A{h}_t+B{x}_t)$ to integrate a sequence and produce an output after a delay. Error correction must now assign responsibility to earlier states as well as earlier layers. Applying the chain rule through the unrolled recurrence solves the algorithmic problem, but storing the required history and transporting a late teaching signal to an earlier synapse is biologically demanding. Repeated multiplication by the recurrent Jacobian matrix can also create the vanishing gradient problem or the exploding gradient problem. A local eligibility trace can retain a record of recent synaptic participation for modulation by a later error signal, although that is not generally an exact replacement for the full recurrent gradient.

A biologically motivated comparison is the cerebellum. A broad input representation can be expanded by granule cells; parallel fibers carry the resulting activity to Purkinje cells, while a climbing fiber provides a separate instructive input. Such a circuit resembles a supervised adaptive filter: an active input is modified in a manner depending on a teaching signal, with Purkinje output affecting downstream motor control. The correspondence is a circuit hypothesis, not a claim that the climbing-fiber event equals the signed scalar $t-y$ in every task.

An instructive experiment recorded Purkinje cells in monkeys tracking a moving visual target. A change in target direction supplied a learning instruction, and subsequent eye velocity and simple-spike activity measured trial-over-trial adaptation. Longer complex spikes were associated with larger subsequent simple-spike depression and stronger behavioural learning. Comparing trials matched for preceding simple-spike activity helped rule out that activity alone as the explanation. This connects a separable teaching-related event to later plasticity during behaviour, but extracellular recordings do not directly reveal the numerical update at every parallel fiber synapse, and the finding does not demonstrate multilayer backpropagation.^[ref]

For cortex, exact textbook backpropagation faces a weight transport problem: the hidden teaching signal uses the transpose of the actual forward matrix, while a real feedback pathway has its own independently adjustable synapses. It also needs a way to deliver a target or prediction error, keep learning signals distinct from ordinary activation, and convert them to local plasticity under spiking, delays and sign constraints. The mere presence of anatomical feedback does not establish this computation.

Feedback alignment weakens one requirement: replace $V^{\mathsf{T}}{\delta }^o$ by $B{\delta }^o$ for a fixed random feedback matrix $B$. Simulations showed that changing forward weights can make these approximate teaching directions useful, even without exact reciprocal weights. This is evidence that the weight transport problem need not forbid useful deep error correction; it is not evidence that arbitrary feedback always converges, that the updates remain exact gradient descent, or that this precise mechanism has been recorded in the brain. The work explicitly leaves the physiological implementation and the interaction between feedback activation and learning unresolved.^[ref]

Local supervised error correction has a plausible circuit substrate and direct behavioural-physiological support. Multilayer and recurrent error correction are powerful explanatory possibilities, but their teaching signals, temporal credit assignment and feedback pathways must be established experimentally. Different architectures should be compared by held-out learning behaviour, representational changes and pathway-specific intervention, rather than by labelling all feedback as backpropagation.

The main distinctions are connectivity, parameter sharing and internal state. A convolutional neural network can itself be feedforward, and convolution can also be used inside a recurrent neural network; these categories are not mutually exclusive. Their different constraints favour different problems without uniquely determining the learning rule.

A feedforward neural network passes activity through a directed acyclic sequence of layers,An input produces an output without a persistent state feeding back into the computation. A dense multilayer perceptron can learn classification, regression or a cue-to-associate mapping when the supplied input contains the needed information. Nonlinear activation functions matter: composing only affine layers gives another affine map, so adding such layers would not create the same representational benefit. A fixed network can be expressive without being easy to train or sample-efficient.

Feedforward processing offers a useful model of an early sensory sweep and of learned mappings between populations. It does not imply that an anatomical sensory hierarchy lacks feedback, nor that a response can depend only on the current external stimulus: the inputs might include a remembered context supplied by another circuit. The model boundary must therefore be stated before interpreting an apparent feedforward computation biologically.

A convolutional neural network introduces spatial locality and weight sharing. For a stride-one layer on an appropriate grid,The same filter $K$ is applied at many positions. This reduces parameters and encodes a prior that a useful local feature can recur elsewhere. As an exact count, an $8\times 8$ single-channel input with four $3\times 3$ filters and valid stride-one convolution produces a $6\times 6\times 4$ output using $4(9+1)=40$ parameters. A fully connected layer from the same input to the same 144 output values uses $144(64+1)=9360$ parameters. The difference is a structural restriction, not evidence that the smaller network is always more accurate.

Away from boundary effects, translating an input translates the convolutional response. This is translation equivariance, not complete invariance. Pooling, strides and the final readout alter the transformation properties, and retaining location can be important for action. Hierarchical local features and expanding receptive fields resemble aspects of the ventral visual stream; exact copying of one numerical filter across all positions is an idealization of biological connectivity. Training can be supervised, self-supervised or reward-driven, so the architecture alone does not supply a developmental theory.

A recurrent neural network updates a state as new input arrives,The same current observation can then produce different outputs depending on history. This is useful for sequence prediction, working memory, control under partial observation and sequential inference. For an original scalar example, let $h_t=h_{t-1}/2+x_t$ with $h_0=0$. Sequences $(1,0)$ and $(0,0)$ have the same final input but final states $1/2$ and $0$. A feedforward map of only that final input cannot distinguish them. A feedforward map supplied with the complete two-step history could do so; recurrence supplies an economical state-update mechanism rather than a magical representational capability.

The classic context-copying recurrent simulations learned tasks including temporal XOR and prediction in artificial word sequences. Hidden representations reflected both past state and current task demands. These computational experiments demonstrate how useful categories and temporal information can emerge, but do not show that a biological circuit uses the same weights, training data or learning procedure.^[ref]

Recurrent learning is also harder to explain biologically. Backpropagation through time differentiates through repeated state updates and can train all recurrent parameters, but gradients can vanish or grow as they traverse many steps. Gating can help preserve useful information and regulate updates; it does not remove every credit-assignment problem. Activity and stability depend on the operating regime. Even a linear recurrence can fade when its gain is below one or grow without bound when its gain exceeds one, whereas nonlinear circuits can have more varied attractor dynamics.

Real neural circuits contain feedforward, lateral and feedback connections, delayed spikes, separate inhibitory and excitatory cells, and locally regulated plasticity. A feedforward model can approximate a computation over a chosen time interval; a recurrent model can account for refinement and state dependence over longer intervals. Neither is a complete cell-level description. Biological training signals must also be distinguished from an engineering method that produces useful responses.

Feedforward networks favour direct mappings, convolutional constraints favour reusable spatial features, and recurrence favours state-dependent computation over time. Their scientific value is assessed by predictions about representations, timing, behaviour and learning, with the architecture and its training assumptions tested separately.

A perceptron is a binary classification unit using an affine score $w^{T}x+b$ and a threshold to predict a label. The perceptron algorithm updates that score online. Absorb the bias into an augmented constant input, use labels $y_i\in \{-1,1\}$, and start at $w=0$. Whenever $y_i{w}^T{x}_i\le 0$, including a tie, updateFor a finite linearly separable dataset, suppose a unit vector $u$ satisfies $y_i{u}^T{x}_i\ge \gamma >0$ and $\Vert x_i\Vert \le R$. After $M$ updates, $u^{T}w\ge M\gamma$, while expanding the squared norm at each update gives $\Vert w{\Vert }^2\le M{R}^2$. The Cauchy-Schwarz inequality yields $M\gamma \le R\sqrt{M}$, hence $M\le (R/\gamma {)}^2$. Cycling through the dataset therefore finds a separator after finitely many updates.

The assumption matters: nonseparable data can keep generating updates. The method finds a separating hyperplane when the assumptions hold, rather than necessarily maximizing its margin as a support vector machine aims to do. A single perceptron also differs from a multilayer perceptron, whose hidden activation functions can represent nonlinear decision regions.