Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 84 (Computational Neuroscience)

A binary Hopfield network has $N$ recurrently connected units $s_i\in \{-1,+1\}$, symmetric weights $w_{ij}=w_{ji}$, zero self-weights, and thresholds ${\theta }_i$. A distributed memory is a desired population pattern ${\xi }^{\mu }$. One Hebbian learning prescription isCoactive signs strengthen a connection, and opposite signs weaken it. This is Hebbian memory weights for a Hopfield network, rather than a separate address for each memory.

Initialize the units with the input cue, which can be a corrupted or partially specified pattern. Update one unit at a time by $s_i\leftarrow \mathrm{sign}({\sum }_j{w}_{ij}{s}_j-{\theta }_i)$, retaining its old sign when the field is exactly zero. A fair asynchronous schedule revisits every unit. The energychanges on a single update by $\Delta E=-(s_i^{\prime }-s_i)({\sum }_j{w}_{ij}{s}_j-{\theta }_i)\le 0$. An actual flip with the tie convention strictly reduces energy; finitely many states imply eventual arrival at a fixed point. This proves asynchronous Hopfield energy descent. Synchronous updating lacks this single-unit argument and can cycle.

For one stored pattern and zero thresholds, the field at $s=\xi$ is $(N-1){\xi }_i/N$, so the pattern is a fixed point. With multiple patterns, the desired signal is accompanied by cross-talk from other memories; not every arbitrary set is guaranteed to be stable. Retrieval is attraction from a cue to a stored-pattern fixed point, provided the cue lies in its basin of attraction. Spurious minima and limited basins are part of the model, not errors in its update algorithm.

To an experimental neuroscientist, a Hopfield network offers a concrete hypothesis about associative memory: information is distributed over a population, a partial cue can complete a learned pattern through recurrent interactions, and synaptic changes can alter which persistent activity patterns are accessible. It predicts attraction basins, interference between similar memories and resilience to some corrupted inputs. These are useful qualitative questions for experiments, and the connection between Hebbian learning, collective dynamics and stored patterns is mathematically testable.

Its limitations are equally concrete. Binary units suppress spike timing, neural adaptation, heterogeneous conductances and dendritic computation. Exact symmetric effective interactions, unrestricted signs and the absence of delays are idealizations; individual biological neurons generally do not implement arbitrary positive and negative outgoing synapses in the way the unrestricted weight matrix permits. The simple learning rule supplies neither realistic forgetting nor a complete account of biological long-term synaptic plasticity. Memory capacity and retrieval reliability depend on load and pattern correlations, and spurious attractors occur. Moreover, a stable activity pattern is not by itself an account of consolidation, behavioral readout or the anatomical location of a memory. It is an explanatory model of associative retrieval, not a literal complete circuit diagram of the brain.

Since the derivative of $\tanh (\beta x)$ at zero is $\beta$, the Jacobian matrix of this directed rate network isWith $a=1+\tau \lambda$, its characteristic determinant isThus the eigenvalues satisfyThis establishes three-neuron ring stability from the loop product. The ring is directed, so the symmetric-weight Hopfield network energy argument does not apply. Assume the physical relaxation time $\tau >0$.

An even number of negative weights gives $\Gamma >0$ when all three connections are nonzero. Put $r={\Gamma }^{1/3}$. The three cube roots giveThe complex pair always has negative real part, and the real eigenvalue is negative exactly when $r<1$. Therefore strict linear asymptotic stability requires $\Gamma <1$. The origin is unstable when $\Gamma >1$. A zero connection gives $\Gamma =0$ and all eigenvalues $-1/\tau$, also stable, though the matrix need not be diagonalizable.

At $\Gamma =1$, linearization has a simple zero eigenvalue, so a strict negative-real-part test alone does not settle nonlinear stability. For completeness, take a nonzero real eigenvector $c$ of the weighted loop matrix $M=\beta W$, satisfying $Mc=c$, and set $x_i=c_i{u}_i$. The local equations areThe center direction has all $u_i=a$. Averaging the three equations on the center manifold gives $\tau \dot{a}=-({\beta }^2{\sum }_i{c}_i^2/9)a^3+O(a^5)$. Its leading coefficient is negative, while the other modes decay exponentially. Hence the exact boundary origin is locally asymptotically stable for the specified hyperbolic-tangent nonlinearity, with algebraic rather than exponential decay. This qualification distinguishes actual nonlinear stability from the usual strict Jacobian criterion.

An odd number of inhibitory connections gives $\Gamma <0$. Write $r=|\Gamma {|}^{1/3}>0$. The cube roots of $-r^3$ have arguments $\pi ,\pi /3,-\pi /3$, soThe real mode is always damped. The pair has negative real part precisely when $r\cos (\pi /3)<1$, givingfor strict linear asymptotic stability. For $r>2$ the origin is unstable to an oscillatory mode.

At $r=2=1/\cos (\pi /3)$ the eigenvalues are $-3/\tau$ and $\pm i\sqrt{3}/\tau$. Thus the linear oscillation frequency is $\boxed{{\omega }_H=\sqrt{3}/\tau }$ and its period is $2\pi \tau /\sqrt{3}$. Varying $r$ crosses the imaginary axis transversely, since $d\mathrm{Re}{\lambda }_{+}/dr=1/(2\tau )>0$.

The hyperbolic tangent has a negative cubic saturation and no quadratic term. To check the nonlinear onset, let $v$ be a critical complex eigenvector of $M=\beta W$ and $p$ its normalized adjoint eigenvector, with $p^{\ast }v=1$. For this weighted cycle, ${\overline{p}}_i{v}_i=1/3$. Since $Mv=(1+i\sqrt{3})v$, the cubic derivative evaluated on $(v,v,\overline{v})$ has projectionIts real part is strictly negative. With no quadratic contributions, this gives a negative first Lyapunov coefficient: the generic onset is a supercritical Hopf bifurcation, with a small attracting periodic orbit just beyond the threshold. Exactly at threshold the linear system has undamped oscillations, but nonlinear cubic damping makes sufficiently small oscillations decay; one must not infer a nonzero sustained cycle at the critical parameter solely from the imaginary eigenvalues.

Apply the chain rule to the squared loss, using the same pre-update weights throughout a forward and backward pass. At the output, $\partial E/\partial z_k=z_k-t_k$ and $\partial z_k/\partial x_k=g_k^{\prime }(x_k)$. ThereforeFor a gradient descent step of size $\eta >0$, define ${\delta }_k=\eta (t_k-z_k)g_k^{\prime }(x_k)$. This gives $\boxed{\Delta w_{jk}={\delta }_k{z}_j}$.

A hidden activation affects all output units to which it connects. ThusandThe learning rate is already contained in ${\delta }_k$ and hence in ${\delta }_j$; do not multiply by it a second time. For a sigmoid function, $g^{\prime }(x)=g(x)[1-g(x)]$. Thresholds can be treated as negative biases with an extra constant input and updated by the same derivative rule.

This is squared-error backpropagation. The forward pass computes activations, while the backward pass transports output-loss derivatives through the outgoing weights and the derivatives of the activation functions, yielding an error signal for every hidden unit. That reverse flow of derivatives explains the name backpropagation.

Nonlinear hidden units create useful intermediate features, allowing the output to combine features rather than classify the original inputs directly. For example, two binary threshold units can compute OR and AND of two binary inputs; thresholding OR minus twice AND at $1/2$ gives XOR. A single monotone threshold of an affine function of the original two inputs cannot implement XOR: the two positive examples lie opposite each other and cannot be separated from the two negative examples by one line. Hidden features can also detect edges or other local patterns before a subsequent layer combines them into object-level features.

If hidden activation functions are identities, substitute $z_j=x_j={\sum }_i{w}_{ij}{z}_i$ into the output input to obtainWith biases included this is still one affine map. Linear hidden layers collapse into a single effective layer and add no nonlinear feature-building ability. The output may remain nonlinear, but its preactivation is just an affine function of the inputs. With standard monotone output activations this still cannot solve XOR or parity by one output unit. An artificially chosen nonmonotone output function could itself encode a complicated scalar decision rule, but that would not be a benefit of the linear hidden layer.

Two arguments favour backpropagation as a modelling tool. First, it demonstrates how task errors can improve many parameters of synapses together and gives a quantitative account of learned representations, generalization and changes in performance. Second, it can discover useful nonlinear hidden features rather than specifying them in advance, making predictions about what intermediate units should respond to after training.

Two arguments oppose interpreting standard backpropagation literally as nervous-system learning. First, the hidden error requires an accurately weighted reverse signal, including the same effective forward weights; this weight-transport requirement is not supplied by the ordinary local Hebbian learning rule. Second, a supplied output target and differentiable layer-by-layer credit assignment idealize the feedback and temporal information available to biological neurons, which use spike trains, delays and local synaptic plasticity. Success at learning a task establishes computational usefulness, not that the nervous system implements the identical algorithm.

Use the four-input parity network. Let $S=z_1+z_2+z_3+z_4$. Four hidden units each receive all inputs with weight one and have thresholds $1/2,3/2,5/2,7/2$. Thuswhere $H(a)=1$ for $a\ge 0$ and zero otherwise. The output receives weights $(1,-1,1,-1)$ and has threshold $1/2$:The computation depends only on input count, so the following table checks all sixteen patterns at once:The output is one exactly for an odd input count. The hidden features detect nested count thresholds, whose alternating sum extracts odd parity.

This representation does not make learning automatically easy. Hard binary threshold units have no usable ordinary derivative; backpropagation normally uses smooth approximations. With uniformly sampled parity patterns, each individual input has zero correlation with the target, so simple first-order features provide no direct cue to the required interaction. The desired output also flips between neighboring count classes. A smooth network has sufficient capacity, but its training can be sensitive to initialization, saturation and local optimization; representability and trainability are different issues.

At rest, the cell has relatively high intracellular potassium and high extracellular sodium. The membrane potential is negative inside, largely because selective resting permeability, particularly potassium leak, combines with these electrochemical gradients. The sodium-potassium pump maintains the gradients over time; it is not the immediate regenerative current of an action potential.

Depolarization rapidly activates sodium voltage-gated ion channels, causing inward sodium current and further depolarization. Slower sodium inactivation and delayed potassium activation then reduce inward current and increase outward potassium current, repolarizing the membrane. Persisting potassium conductance can produce an after-hyperpolarization; recovery of sodium availability contributes to the neuronal refractory period. A spike transfers only a small fraction of the bulk ions, so it changes charge separation much more than it changes the overall concentration gradients.

For two-state ion-channel gating, let $X$ be the fraction of permissive gates. Opening transfers probability from $1-X$ at rate ${\alpha }_X(V)$, and closing transfers it back at rate ${\beta }_X(V)$:Independent subunits that must all be permissive multiply their probabilities. The Hodgkin-Huxley model therefore uses $g_{\mathrm{Na}}={\bar{g}}_{\mathrm{Na}}{m}^{3}h$ and $g_{\mathrm{K}}={\bar{g}}_{\mathrm{K}}{n}^4$: $m$ describes sodium activation, $h$ sodium availability, and $n$ potassium activation. Their voltage-dependent time scales produce the sequence of inward and outward current. This channel-level construction is the mechanism in the primary Hodgkin–Huxley membrane-current model.

A dynamic clamp is a real-time feedback experiment. Measure the membrane potential through an electrode, numerically evolve chosen channel or synapse state variables, calculate the modeled current, and inject that current into the real neuron. A virtual synapse might use $I_{\mathrm{syn}}=g_{\mathrm{syn}}(t)(E_{\mathrm{syn}}-V(t))$; a voltage-dependent virtual channel also updates its gate probabilities from the measured voltage. Repeat at a rate fast relative to the dynamics being modeled. Unlike voltage clamp, this procedure does not hold the voltage at a command value.

Two uses are to add a controlled inhibitory conductance and measure its effect on firing or integration, and to couple two real neurons by artificial reciprocal inhibitory synapses to test the resulting network dynamics. Both artificial conductances and reciprocal inhibitory connections were demonstrated in the primary Sharp–O'Neil–Abbott–Marder dynamic-clamp experiments. Conductance magnitude, time course and reversal voltage can be varied without requiring the cell to express a new channel. Sampling delay, electrode error and the inability of a single injection site to reproduce all dendritic conductance locations are practical limitations.

The leaky integrate-and-fire model treats the membrane as a capacitance $C_m$. The leak conductance $g_L$ drives voltage towards its resting reversal $V_L$; the excitatory conductance $g_E$ drives it towards its reversal $V_E$; and $I_e$ is injected current, positive inward. A sufficiently high $V_E$ makes increasing $g_E$ depolarizing over the relevant voltage range, while also increasing total conductance and shortening the integration time.

For constant input and conductances, putThenIf this trajectory reaches $V_{\theta }$, register a spike and reset the voltage to $V_0<V_{\theta }$, optionally enforcing a specified neuronal refractory period. The reset makes repeated threshold crossings possible under a constant suprathreshold drive. If $V_{\infty }\le V_{\theta }$ and the initial voltage is below threshold, no finite-time crossing occurs; equality gives asymptotic approach. For time-varying currents or conductances, integrate the subthreshold differential equation until the next crossing.

Use this model when spike timing, spike counts and network integration matter more than the detailed waveform: it is inexpensive for large recurrent networks or comparisons of current-driven and conductance-driven input. It omits ionic spike initiation, channel-specific adaptation and many dendritic effects, for which a Hodgkin-Huxley model or a spatial model may be needed.

Assume constant current, $g_L>0$, $C_m>0$, and reset below threshold. With $g_E=0$, $V_{\infty }=V_L+I_e/g_L$. A spike occurs in finite time after reset precisely when $V_{\infty }>V_{\theta }$, so the critical boundary isAt equality the voltage approaches threshold asymptotically, so the finite-time firing condition is the strict inequality $I_e>I_c$.

Solve the exponential voltage trajectory from reset to threshold. The interspike interval of this constant-input firing rate of a leaky integrate-and-fire neuron isThusThe logarithm decreases strictly with current, so $r$ increases, starting from zero as $I_e\downarrow I_c$. At large current, its argument increment is small andThe second expression states the leading linear growth, not an exact intercept formula. If an explicit refractory time $t_{\mathrm{ref}}$ is added, the rate instead becomes $1/(T+t_{\mathrm{ref}})$ and approaches $1/t_{\mathrm{ref}}$ at very high current. The printed model contains no such extra interval, which is why its unrestricted high-current rate is linear.

Fit a stimulus-conditioned linear-nonlinear-Poisson cascade model. Choose a causal stimulus-history window, subtract the stimulus mean, and estimate a temporal filter $k$. A useful initial estimate is the spike-triggered average, $\hat{a}(u)=N^{-1}{\sum }_i[s(t_i-u)-\bar{s}]$. For white Gaussian stimulation and a one-filter response, this estimates the filter direction. With correlated Gaussian stimulation it estimates a direction proportional to $Ck$, where $C$ is the history covariance matrix; use stimulus-correlated spike-triggered averaging to whiten or regularize this correction. An arbitrary stimulus or an even response nonlinearity does not guarantee that this average identifies the correct model.

A general fitting approach is to expand a causal filter in smooth basis functions, compute $a(t)=b+{\int }_0^{U}k(u)s(t-u)du$, and take a nonnegative rate $\lambda (t)=f(a(t))$. With $f(a)=e^a$, fit the coefficients by Poisson spike-train likelihood fitting:The first term rewards rate at observed spikes and the second penalizes overprediction elsewhere. Include a filter-size or smoothness penalty, and choose complexity on held-out data. Another option estimates $f$ by stimulus-projection histograms: divide the density of projection values at spikes by their occupancy density, multiplied by the mean firing rate. Avoid interpreting poorly sampled bins as reliable rates. Temporal train/test splits should avoid leakage through the filter-history window.

Given the fitted rate and the same stimulus, generate an Inhomogeneous Poisson process. For small bins of width $\Delta t$, spike with probability $1-e^{-\lambda (t)\Delta t}$, choosing the bins sufficiently short that multiple events are negligible. More exactly, advance integrated intensity until it increases by an independent unit-exponential random variable, then place the next event. Repeating this generates different synthetic spike trains with the same fitted stimulus-dependent rate, not copies of the original event times.

Compare held-out log-likelihood, stimulus-conditioned firing-rate predictions, count distributions, interspike-interval distributions and autocorrelation. With repeated identical stimulus presentations, compare trial-averaged time courses and trial-to-trial count variance, including the Fano factor. A single recorded train does not by itself supply a reliable repeated-trial average or variance; use held-out temporal data, model-based uncertainty and further recordings instead. Time-rescaled spike-train diagnostics test whether integrated-intensity intervals look independently exponential, rather than comparing raw intervals to an exponential law under a changing rate.

A successful model reproduces specified statistical features on new data; identical spike times are not the goal. The basic Poisson model can miss refractoriness, bursting, adaptation, history dependence or stimulus-unexplained trial variability. If needed, add a spike-history filter to the conditional rate, including an absolute dead time, and fit the resulting history-dependent model. A complex cell may also require multiple filters or quadratic stimulus features: its spike-triggered mean can vanish despite strong tuning. These failures identify model limitations rather than evidence that a random spike generator should match every observed interval.

A complex cell in primary visual cortex typically responds to a suitably oriented bar, edge or grating over a finite receptive-field region, with less dependence on precise spatial phase and contrast polarity than a simple cell. Its response need not reveal separated linear ON and OFF subregions. Many cells have additional motion or binocular selectivity, so phase tolerance alone is not a complete account of every complex cell. Moving the feature beyond its receptive field still reduces the response.

In a pooling model of a complex cell, several simple cells with similar preferred orientation but different positions or phases feed the output. For linear subunit responses $a_m=⟨k_m,I⟩$, a possible model is $r=f({\sum }_m{w}_m[a_m-{\theta }_m{]}_{+})$, with $w_m\ge 0$. Include opposite-polarity filters if both light and dark features are to drive the cell. Rectification stops cancellation between subunits, and pooling spreads position and phase sensitivity across their coverage.

In a quadrature energy model of a complex cell, use two oriented filters of matching frequency but quadrature phase, and setFor a matched grating, choose the pair so $a_e=A\cos \phi$, $a_o=A\sin \phi$; then $E=A^2$, independent of grating phase while retaining orientation and frequency tuning. Finite approximate filter pairs give only approximate invariance for general images. The quadrature construction underlies the primary Adelson–Bergen energy-model work; temporal filters can extend it to motion sensitivity.

The pooling model emphasizes a circuit of rectified subunits and can flexibly cover translated positions. The energy model gives an explicit algebraic phase-invariance mechanism with quadratic contrast dependence before its output nonlinearity. They are not mutually exclusive biological alternatives: suitable subunit pooling can approximate an energy computation. Their spatial tolerance, contrast response and direction selectivity depend on the chosen filters and nonlinearities, and both require comparison with measured receptive fields.

Theoretical models make developmental explanations constructive. Rather than assigning every final receptive field and cortical map in advance, build a network with coarse anatomical constraints, initially weak or imperfect connections, activity statistics and a synaptic plasticity rule. Ask whether the observed organization emerges, what sets its spatial scale, and which perturbations alter it. This distinguishes a mechanism capable of producing a pattern from a descriptive picture of that pattern.

A common starting point for correlation-based development of visual cortical maps is a constrained Hebbian learning rule. If $z_i$ is an afferent activity and $y_a$ a cortical response, write ${\dot{w}}_{ai}=ϵ⟨y_a{z}_i⟩-{\Lambda }_a$, together with bounds on the weights. The normalization term ${\Lambda }_a$ can subtract the mean update to hold total incoming weight fixed; other models rescale its norm. Without synaptic normalization or competition, Hebbian positive feedback can simply make every correlated connection grow. For linear activity $y_a={\sum }_{b,j}{K}_{ab}{w}_{bj}{z}_j$, averaging gives $⟨y_a{z}_i⟩={\sum }_{b,j}{K}_{ab}{w}_{bj}{C}_{ji}$. This shows directly how input correlations, lateral interactions and constraints select growing modes of connectivity.

One example is activity-dependent ocular dominance segregation. Initially overlapping left- and right-eye afferents compete for cortical targets. Correlation within each eye and weaker correlation across eyes can amplify an eye-preference difference. In a simplified local two-eye model, the linear operator on $d_a=w_{aL}-w_{aR}$ contains $K_{ab}(C_{\mathrm{same}}-C_{\mathrm{cross}}{)}_{ab}$. Spatial eigenmodes reveal whether a binocular state is unstable to segregation; saturation prevents unlimited difference growth. The primary Miller–Keller–Stryker model linked correlation structure and intracortical interactions to monocular organization and column width. Its importance was the testable dependence of organization on measurable parameters, rather than just producing alternating stripes.

A second example is orientation-map self-organization. Spatially arranged ON and OFF afferents provide candidate inputs to cortical cells. Correlation-based plasticity with competitive constraints can strengthen subsets that form elongated simple cell receptive fields; lateral interactions coordinate preferences among nearby cells. This makes the development of orientation selectivity and the arrangement of preferences across cortex related but distinct questions. Classic work includes von der Malsburg's self-organization model, while Erwin–Miller correlation-based development models tested requirements on activity for jointly organized orientation and eye preference. Changing the input correlation structure or competition rule provides a concrete way to test which ingredients matter.

These networks motivate deprivation or altered-input simulations: weakening one eye's effective drive can favor the other under competition, while changing correlations can alter segregation or selectivity even without simply reducing mean rate. Such outcomes generate experimental hypotheses; they are not universal consequences of every Hebbian rule. Activity before patterned vision can also enter these models as structured spontaneous input, so activity-dependent development is not synonymous with learning from viewed images.

The main contribution is a quantitative bridge from plasticity and activity statistics to receptive fields and cortical organization. The models also expose what remains unexplained. Molecular guidance, arbor geometry, inhibition, homeostasis, species differences and developmental timing constrain which networks are plausible. Similar final maps can arise from different learning mechanisms, so agreement with a map alone does not identify the actual synaptic rule. Stronger conclusions require successful predictions of how organization changes under controlled perturbations, not merely resemblance to one mature cortical pattern.