SYDE 552: Computational Neuroscience

Synapses are not static. The Tsodyks–Markram model treats each terminal as having a finite resource \(x \in [0,1]\) of releasable vesicles and a utilization variable \(u\) that rises with activity and decays slowly; spikes consume \(u x\) of the resource and cause \(u\) to increment. With different parameter settings this yields short-term depression (the dominant behaviour at excitatory cortical synapses onto pyramidal cells), short-term facilitation (at pyramidal-to-interneuron synapses), or both. Short-term plasticity implements a dynamic gain control that emphasizes novel or transient inputs.

5. Rate models and mean-field dynamics

For many questions the precise spike times are dispensable and one can track only the instantaneous firing rate \(r_i(t)\) of each population \(i\). Under the relaxation-to-f-I assumption,

with \(F_i\) a sigmoid or rectified-linear nonlinearity. Rate models are the natural bridge between biophysics and machine learning: they are deterministic ODEs that can be trained by backpropagation through time.

5.1 Wilson–Cowan equations

For a single cortical column with one excitatory and one inhibitory population,

Depending on parameters the system has a single low-activity fixed point, bistability between low and high activity, sustained oscillations (a limit cycle arising from a Hopf bifurcation), or more exotic behaviour. Oscillations in the \(\gamma\)-band (30–80 Hz) are a natural explanation of cortical rhythms and have been used to model attentional modulation.

5.2 Linear stability and rhythms

Linearizing around a fixed point and computing the eigenvalues of the Jacobian determines whether perturbations decay, grow, or oscillate. A negative real part gives stability; imaginary parts give damped oscillations whose frequency is set by the cross-coupling strengths and time constants. This analysis explains why pyramidal-interneuron networks with delays are natural oscillators — the so-called PING and ING mechanisms.

6. Neural coding

6.1 Rate versus timing

A long-running debate concerns whether cortex uses mean firing rate or precise spike timing to carry information. The evidence is mixed and domain-specific. Retinal ganglion cells and auditory nerve fibres exhibit sub-millisecond temporal precision; cortical neurons encode slow analogue variables (luminance, contrast, orientation) primarily in firing rate, but also carry information in oscillatory phase and cross-neuron synchrony. Rather than picking a side, the modern view is that time scales of stimulus statistics determine the relevant code: rapid transients demand temporal codes, slow variables tolerate rate codes.

6.2 Population coding

Single neurons are noisy and tuned to a narrow range of stimuli; populations overcome both limitations. If \(N\) neurons have tuning curves \(r_i(\theta)\) and independent Poisson noise, the Fisher information about \(\theta\) is

The Cramér–Rao bound tells us that no unbiased estimator of \(\theta\) achieves lower variance than \(1/I_F\). In well-behaved cases this bound is tight and the population acts as a near-optimal encoder. Correlated noise shared across neurons can reduce \(I_F\) — so-called information-limiting correlations — or leave it unchanged, depending on whether the noise lies along the signal direction in population-response space.

6.3 Decoding

Several canonical decoders are used. The population vector (Georgopoulos) sums the preferred directions of each neuron weighted by its firing rate and is optimal only for very specific cosine tuning. Maximum-likelihood decoding chooses \(\hat\theta = \arg\max_\theta \prod_i p(r_i \mid \theta)\) and asymptotically saturates the Cramér–Rao bound. Bayesian decoding combines the likelihood with a prior \(p(\theta)\) and returns the full posterior, which is essential when modelling behavioural biases or Bayes-optimal inference from uncertain inputs.

7. Receptive fields and early vision

7.1 Linear receptive fields

The receptive field of a visual neuron is the spatiotemporal stimulus kernel that best predicts its response. For retinal ganglion cells and LGN relay neurons the receptive field is well modelled by a centre-surround difference of Gaussians in space and a biphasic temporal kernel, producing band-pass filtering in both domains. Spike-triggered averaging — the mean stimulus preceding a spike — estimates this linear kernel, and spike-triggered covariance recovers higher-order subunits.

7.2 Simple and complex cells in V1

Hubel and Wiesel’s classic result is that primary visual cortex contains at least two classes of orientation-selective neurons. Simple cells respond linearly to oriented spatial patterns and are well fit by Gabor functions of the form

where \((x',y')\) is a rotated coordinate system. Complex cells retain orientation tuning but are invariant to the position and phase of the grating within their receptive field; they are well modelled as a sum of squared simple-cell responses, an early instance of the quadrature pair construction that motivates hierarchical pooling in convolutional networks. An “energy model” formed by squaring and summing two Gabor-filtered responses in quadrature produces phase-invariant orientation tuning identical to that of complex cells.

7.3 Natural image statistics

Why Gabors? Olshausen and Field showed that if one demands a linear generative model of natural images in which only a small fraction of basis vectors is active per image — sparse coding — the learned basis resembles oriented, band-pass, localized filters, that is, Gabors. Independent component analysis, applied to whitened natural image patches, recovers the same basis. This provides a normative explanation of V1 receptive fields: they are a near-optimal dictionary for representing the visual world efficiently.

8. Synaptic plasticity I: Hebbian and rate-based rules

8.1 Hebb’s postulate

Hebb’s 1949 postulate — “cells that fire together, wire together” — becomes, in its simplest mathematical form,

This rule extracts the first principal component of the input correlation matrix in its fixed points and explains, qualitatively, the development of orientation selectivity from correlated retinal inputs. It is unstable: weights grow without bound unless constrained.

8.2 Oja’s rule and PCA

Oja’s modification,

adds a weight-dependent decay that normalizes \(\|w\|\) and causes the weight vector to converge exactly to the leading eigenvector of the input covariance matrix. Stacks of Oja-learning units recover successive principal components. This makes the rule normatively interpretable: Hebbian learning under a specific normalization implements online PCA.

8.3 BCM theory

Bienenstock, Cooper, and Munro addressed a different problem: Hebbian rules alone cannot produce selectivity because there is no mechanism to depress rarely active synapses. The BCM rule introduces a sliding threshold \(\theta_M\) that depends on the running average of postsynaptic activity:

Synapses potentiate when the postsynaptic rate exceeds \(\theta_M\) and depress when it falls below. The quadratic moving threshold provides homeostasis: a highly active neuron raises its threshold, making further potentiation harder. BCM reproduces ocular dominance plasticity in visual cortex and provides a rate-based abstraction of LTP/LTD.

9. Synaptic plasticity II: STDP and molecular substrates

9.1 Spike-timing-dependent plasticity

Recordings by Bi and Poo, and by Markram, showed that the sign and magnitude of synaptic change depend on the relative timing of pre- and postsynaptic spikes. The canonical additive STDP window is

with \(\Delta t = t_{\text{post}} - t_{\text{pre}}\) and time constants of 10–30 ms. Pre-before-post produces LTP, post-before-pre produces LTD. STDP is a temporal generalization of Hebb’s rule that captures the causal asymmetry of information flow. With appropriate parameter choices STDP performs a kind of unsupervised feature extraction and, in the limit of slow, low-rate firing, reduces to a BCM-like rule.

9.2 LTP, LTD, and NMDA

At the molecular level, long-term potentiation at many glutamatergic synapses depends on NMDA receptor activation. NMDA receptors conduct calcium only when the membrane is depolarized (relieving the magnesium block) and glutamate is bound — they are biochemical AND-gates for coincident pre- and postsynaptic activity. Brief, large calcium influx recruits CaMKII-mediated signalling cascades that insert AMPA receptors, producing LTP. Prolonged, modest calcium influx preferentially activates calcineurin, causing AMPA removal and LTD. The bidirectional dependence on postsynaptic calcium concentration is the biophysical foundation for the BCM sliding threshold.

9.3 Homeostatic plasticity

Hebbian rules and STDP are inherently positive-feedback processes. Homeostatic mechanisms — synaptic scaling, intrinsic excitability adjustments, and metaplasticity of plasticity thresholds — counteract runaway dynamics by adjusting whole-cell gain on slow timescales (hours to days). Stable networks require both fast Hebbian learning to extract structure and slow homeostasis to prevent saturation.

10. Recurrent networks and attractor dynamics

10.1 Hopfield networks

Hopfield’s binary recurrent network stores patterns as fixed-point attractors. With \(N\) binary units and weights

each stored pattern \(\xi^\mu\) is a local minimum of the Lyapunov energy \(E = -\tfrac{1}{2}\sum_{ij}W_{ij}s_i s_j\) provided the patterns are approximately orthogonal and \(P \lesssim 0.138 N\). Above this critical capacity the network transitions to a spin-glass phase of spurious attractors. Continuous-valued Hopfield networks with a differentiable energy function are the ancestor of modern energy-based models, and modern Hopfield networks with softmax updates have been shown by Ramsauer et al. to be mathematically equivalent to the attention mechanism of transformers.

10.2 Continuous attractors

Head-direction cells in the rodent limbic system, place cells in hippocampus, and oculomotor integrator neurons behave as if they live on a one- or two-dimensional continuous manifold of stable states. Ring attractor networks — recurrent networks with Mexican-hat connectivity — realize such manifolds. An external input nudges the bump of activity along the ring and, in the absence of input, the bump persists, implementing working memory and path integration.

10.3 Line attractors and neural integrators

A line attractor is a marginally stable one-dimensional manifold used as an analogue integrator. The oculomotor integrator in the brainstem is the classical example: it integrates velocity commands to produce eye position. Theoretical analysis shows that the integrator requires fine-tuning of recurrent weights to achieve the required eigenvalue of exactly one along the integrator direction, a tuning problem partially solved by homeostatic mechanisms.

10.4 Chaos in random networks

Sompolinsky, Crisanti, and Sommers showed that a random recurrent network of rate neurons with weights \(W_{ij} \sim \mathcal N(0, g^2/N)\) undergoes a transition from a silent fixed point to spontaneous chaotic activity as \(g\) exceeds unity. Using a dynamical mean-field theory one can derive the autocorrelation structure of the chaotic regime and show that it possesses a broad spectrum of timescales. Networks operating at the edge of chaos are candidate substrates for the echo-state and reservoir-computing paradigms: the rich intrinsic dynamics provide a high-dimensional temporal basis that a simple readout can use to approximate arbitrary input-output maps.

10.5 Balanced networks

Cortical neurons receive thousands of excitatory and inhibitory inputs whose sum is, on average, close to threshold. Van Vreeswijk and Sompolinsky’s balanced-network theory shows that large recurrent networks with strong, random excitation and inhibition spontaneously self-organize into a regime where excitation and inhibition cancel on average, leaving firing driven by fluctuations. This naturally produces irregular, Poisson-like firing and linear population input-output curves, reconciling the high variability of cortical spike trains with deterministic neural dynamics.

11. Reinforcement learning and dopamine

11.1 Classical conditioning and Rescorla–Wagner

Animal learning theorists long before computational neuroscience had a predictive-error rule: the change in associative strength between a conditioned stimulus (CS) and an unconditioned stimulus (US) is proportional to the difference between the actual US and the US predicted by all concurrent cues,

This is Rescorla–Wagner, and it accounts for blocking, overshadowing, and conditioned inhibition.

11.2 Temporal-difference learning

Sutton’s temporal-difference (TD) algorithm generalizes Rescorla–Wagner from episodic to trial-by-trial temporal settings. It updates a value function \(V(s_t)\) using the bootstrapped error

with \(\gamma\) a discount factor. Schultz, Dayan, and Montague’s landmark result is that the phasic firing of midbrain dopaminergic neurons in the ventral tegmental area and substantia nigra pars compacta closely tracks \(\delta_t\): dopamine rises when reward exceeds expectation, is silent when reward matches expectation, and dips below baseline when expected reward is omitted. Dopamine is, to a first approximation, a broadcast reward-prediction-error signal.

11.3 Actor-critic architectures and the striatum

TD-based reinforcement learning is naturally decomposed into a critic that learns \(V(s)\) and an actor that learns a policy \(\pi(a \mid s)\). A standard neural mapping places the critic in ventral striatum and the actor in dorsal striatum, with dopaminergic prediction errors modulating corticostriatal synapses in both structures via three-factor plasticity rules of the form

These rules solve the temporal credit assignment problem by combining slow-decaying eligibility traces at each synapse with a globally broadcast scalar reinforcement signal.

12. Dynamical mean-field theory and random matrix theory

Large random networks are analytically tractable because many quantities self-average in the limit \(N \to \infty\). Dynamical mean-field theory replaces the inputs to a representative neuron by a Gaussian process whose autocorrelation is determined self-consistently. For the Sompolinsky–Crisanti–Sommers network the autocorrelation \(C(\tau) = \langle x_i(t)x_i(t+\tau)\rangle\) obeys a differential equation that can be solved by quadrature, revealing the chaotic phase and the nature of the transition. Random matrix theory, in turn, predicts that the eigenvalues of a random Gaussian \(N\times N\) connectivity matrix fill a disk of radius \(g\) in the complex plane (the circular law). Loss of stability occurs when this disk pushes past the unit circle, a result that explains the critical coupling \(g=1\).

13. Cortical circuits and excitation–inhibition balance

A recurring finding in electrophysiology is that excitatory and inhibitory inputs to cortical neurons co-vary on millisecond timescales: inhibition tracks excitation with a short delay, cancelling most of it and leaving a fluctuation-driven residual that causes spiking. This detailed balance is thought to underlie the linear, fast input-output transformations of cortex, the stability of recurrent dynamics in the face of strong feedback, and the ability of cortical networks to support both reliable averaged responses and rich variability. Several theoretical frameworks — balanced networks, stabilized supralinear networks (SSNs), and inhibition-stabilized networks (ISNs) — make these ideas precise. The SSN model of Ahmadian, Rubin, and Miller explains surround suppression, normalization, and contrast invariance in V1 through a single recurrent mechanism with supralinear input-output curves.

14. Deep learning through a neuroscientific lens

14.1 Convolutional networks and the ventral stream

The ventral visual stream runs retina → LGN → V1 → V2 → V4 → IT. Each stage exhibits progressively larger receptive fields, more complex tuning, and greater position and scale invariance. This is the hierarchy that Fukushima’s Neocognitron and LeCun’s convolutional networks were designed to imitate: alternating convolutional (Gabor-like, simple-cell-like) and pooling (complex-cell-like) layers produce representations that become more invariant and more abstract with depth.

The experimental validation of this analogy is the Yamins–DiCarlo result. Deep convolutional networks trained on large-scale image classification develop internal representations that linearly predict the activity of neurons in V4 and IT better than any previous model, and models that perform better on ImageNet also predict neural activity better — up to a point. The hierarchy of layers in a deep CNN aligns, at least in linear-decodability terms, with the hierarchy of ventral stream areas: early layers predict V1 and V2, middle layers predict V4, late layers predict IT. This is one of the most striking convergences between engineered and biological vision.

14.2 Recurrent networks and cortex

Recurrent networks in machine learning — LSTMs, GRUs, and more recently transformer-based sequence models — solve temporal credit-assignment problems that cortical networks must also solve. Task-trained RNNs reveal circuit solutions that can be compared with electrophysiological data. Mante, Sussillo, and colleagues trained RNNs on a context-dependent decision task performed by monkeys and found that the trained network and the prefrontal cortex both implement the same solution: a low-dimensional dynamical system with selection vectors that route different input features to a shared decision axis. This style of model-mediated reverse engineering has become a dominant paradigm in systems neuroscience.

14.3 Normalization

Divisive normalization — where a neuron’s response is divided by a weighted sum of nearby responses — is ubiquitous in cortex and appears in every modern deep network under various names (batch norm, layer norm, group norm, local response norm). Heeger’s normalization model of V1 explains contrast-response functions, cross-orientation suppression, and surround suppression through a single divisive stage. That deep networks required similar mechanisms to be trainable at scale is a striking convergence.

15. Biologically plausible learning

15.1 The credit assignment problem

Backpropagation solves credit assignment by computing, for every weight, the gradient of a global loss. This requires propagating error signals backwards through the exact transpose of the forward weights, a symmetry the brain has no obvious way to enforce, and it requires a separate backward pass that is not observed experimentally. These are the core reasons the brain is generally assumed not to implement literal backpropagation.

15.2 Feedback alignment

Lillicrap and colleagues showed that if the backward pass uses random fixed matrices instead of the transposes of the forward weights, deep networks still learn — the forward weights progressively align so that their transposes resemble the random feedback matrices. Feedback alignment relaxes the symmetry requirement of backprop and is much more biologically plausible, though it is less effective on very deep convolutional networks.

15.3 Predictive coding

Rao and Ballard’s predictive coding theory treats cortex as a hierarchical generative model. Each area generates a prediction of the activity of the area below; only the prediction error is transmitted upward, and each area’s state minimizes its own prediction error. The algorithm can be implemented with local Hebbian-like plasticity and produces responses consistent with extra-classical receptive field effects (end-stopping, surround suppression) as natural consequences of explaining away. Whittington and Bogacz proved that certain predictive-coding networks implement backpropagation-equivalent credit assignment under weak assumptions, bridging the plausibility gap.

15.4 Three-factor rules and neuromodulation

Local synaptic rules can approximate gradient descent when combined with a globally broadcast third factor — a dopamine-like scalar signal. Rules of the form

where \(\tilde r\) are low-pass-filtered activity traces and \(M(t)\) is a neuromodulatory signal carrying reward, novelty, or error information, can approximate a wide range of supervised and reinforcement objectives. They represent a plausible interface between biological plasticity and the supervised gradients that dominate machine learning.

15.5 Equilibrium propagation and energy-based learning

In energy-based models the network state minimizes an energy function; gentle perturbation of the output toward the target and a second minimization produce, by comparing the two equilibria, gradients that coincide with backprop in the small-perturbation limit. Equilibrium propagation provides a mathematically clean, biologically feasible scheme for learning in recurrent networks and has stimulated renewed interest in Hopfield-style dynamics as substrates for learning rather than merely memory.

16. Differences between brains and artificial neural networks

Several structural asymmetries between biological and artificial neural networks remain unresolved.

Energy. The human brain operates on roughly 20 W. A large modern neural network trained on tensor hardware consumes kilowatts during training and watts at inference for comparable tasks. The brain’s efficiency is thought to reflect sparse coding — only a small fraction of neurons are active at any moment — and event-driven asynchronous spiking communication, which neuromorphic hardware such as Intel’s Loihi and SpiNNaker attempts to emulate.

Sparsity. Cortical firing rates are low (typically 1–10 Hz average, with silent neurons common) and representations are sparse, with only a few percent of neurons strongly active for any given stimulus. Artificial networks with ReLU nonlinearities achieve modest sparsity through activation thresholding, but deliberate sparsity-inducing objectives (Olshausen–Field sparse coding, L1 regularization) remain an area of active research.

Dale’s principle. Biological neurons release only excitatory or only inhibitory neurotransmitters, constraining the sign of their outgoing connections. Artificial networks have no such constraint. Respecting Dale’s principle changes the structure of trainable dynamics and forces different solutions to the same tasks.

Temporal dynamics. Biological synapses have rich short- and long-term dynamics; artificial synapses are typically scalar weights updated by SGD. Adding biophysical realism to artificial networks tends to slow learning but can improve temporal credit assignment in sequence tasks.

Architecture search. Evolution and development have tuned cortical architecture over hundreds of millions of years; artificial architectures are hand-designed or searched over hours. The implicit priors encoded in cortical architecture — specific inhibitory cell types, laminar structure, specific long-range projections — are largely missing from artificial networks.

Credit assignment. Already discussed: the brain lacks a clear substrate for backpropagation, and the extent to which biologically plausible alternatives can close the performance gap with backprop on large tasks is an open research question.

17. Putting it together

A useful mental map of the field is as follows. At the bottom sits the biophysics of single neurons — Hodgkin–Huxley, the cable equation, and reductions thereof — which determines the computational primitives available to the system. Above this sit network dynamics, where recurrent interactions among populations produce oscillations, attractors, and chaotic regimes that can be analysed by linearization and mean-field methods. Sitting on top of both is the plasticity layer, where Hebbian, STDP, BCM, and reward-modulated rules modify synapses so that networks come to represent useful quantities. Comparing this stack to the stack of deep learning — linear layers and nonlinearities, backprop-trained recurrent dynamics, and gradient-based plasticity — reveals both parallels and asymmetries. The parallels (hierarchical feature learning, normalization, attractor-like representations) motivate using deep networks as models of the brain. The asymmetries (biological plausibility of learning rules, energy efficiency, sparse coding, temporal dynamics) define the frontier of the field.

18. Reference table of models and scales

The quantities \(\tau_m\), \(\lvert E_K\rvert\), \(\lvert E_{\mathrm{Na}}\rvert\), and \(\lvert V_{\text{th}} - E_L\rvert\) are the four parameters that, to first approximation, set the timescale and amplitude of every phenomenon discussed above; internalizing their numerical values is the single most useful piece of dimensional intuition for the field.

19. What to take away

Three themes recur across every topic in this course. First, neural systems are fundamentally dynamical: their behaviour is best understood by writing down differential equations, locating fixed points, and analysing stability. Second, learning is plasticity plus an objective: Hebbian rules implement unsupervised correlation extraction, BCM and STDP add homeostatic selectivity, and neuromodulated three-factor rules implement reinforcement learning. Third, representations are the bridge between neuroscience and machine learning: the same ideas — sparse coding, normalization, hierarchical feature extraction, attractor dynamics — appear on both sides of the divide, and each translation in either direction has proven fruitful. The student who leaves this course able to write down a network, analyse its dynamics, attach a plausible plasticity rule, and compare the result with both electrophysiological data and a corresponding deep-learning system has the core competency the field rewards.