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An associative memory retrieves a stored pattern from a related cue. An incomplete cue supplies only some coordinates; a corrupted cue supplies some wrong coordinates. A Hopfield network implements retrieval by making intended patterns stable states with surrounding basins of attraction. This turns correction into a collective computation: no unit needs to know the entire desired answer. A cue outside the appropriate basin of attraction can retrieve another pattern or an unintended stable state.

Suppose a compact connected component $C$ of $\{x:V(x)\le c\}$ contains an equilibrium $x_{\ast }$ and $\dot{V}<0$ throughout $C\setminus \{x_{\ast }\}$. Then $C$ is an invariant sublevel set, and every trajectory in $C$ converges to $x_{\ast }$ by the LaSalle invariance principle. Thus the interior of $C$ lies in the basin of attraction of $x_{\ast }$. The largest such component supplies the strongest basin estimate available from that Lyapunov function alone.

The butterfly effect is the intuitive idea that an arbitrarily small change in an initial condition can eventually change the observed future substantially. In dynamical systems, sensitive dependence makes this precise for a map $F:X\to X$ on a metric space $(X,d)$: there is one separation scale $\delta >0$ such thatThe same $\delta$ works at every point, whereas $y$ and the separation time $n$ may depend on the point and the requested closeness. For a flow map, replace the iteration of a map $F^n$ by ${\Phi }_t$ with some $t\ge 0$. The state space matters: restricting to a single fixed point removes the alternative nearby states needed for sensitivity.

Sensitivity is compatible with deterministic evolution. Given an exact initial condition, there is still one orbit; uncertainty concerns which exact state a measurement represents. It also allows accurate short-term prediction. For a continuous function $F$, each fixed finite iterated function $F^n$ is continuous, so sufficiently close initial states remain close at any specified finite step. What fails is a single initial accuracy that controls every future step.

The logistic map illustrates both sides. At $r=2.8$, nearby orbits in the basin of attraction of its attracting fixed point approach one another. At $r=4$, arbitrarily close states can eventually separate by a macroscopic fraction of $[0,1]$. The definition asserts the existence of separating neighbours, not that every nearby pair must separate: $x$ and $1-x$ have exactly the same next state under the logistic map and hence the same subsequent orbit. Nor must separation increase at every step.

A positive Lyapunov exponent describes average exponential growth of an infinitesimal displacement along an orbit, when the relevant limit exists. It is related to sensitivity but is not a substitute for its uniform definition. For a differentiable interval map, the chain rule givesAt $r=4$, the logistic map has $\lambda =\log 2$ for Lebesgue measure-almost every initial condition, while exceptional orbits can have different exponents. Finite-separation growth is only approximately exponential while the linear approximation remains useful. In a bounded state space, a separation cannot grow exponentially forever.

Sensitivity alone is also weaker than Devaney chaos, which additionally requires topological transitivity and dense periodic points. For example, $F(x)=2x$ on the real line has sensitive dependence and a positive Lyapunov exponent, but its only periodic point is zero, so it is not Devaney-chaotic. This distinguishes growth of uncertainty from the richer organization of recurrent orbits.

Edward N. Lorenz's "Deterministic Nonperiodic Flow" (1963) studied a thermal convection model with three ordinary differential equations. Its numerical orbits exhibited persistent nonperiodic motion and instability to small changes of the initial condition. Lorenz discussed the consequences for very-long-range weather forecasting: a deterministic law need not make a detailed distant future reliably recoverable from approximate observations. The historically relevant lesson of the Lorenz system is this growth of uncertainty under deterministic evolution.^[ref]

A useful hippocampal episodic memory model assigns complementary computations to competitive learning, recurrent autoassociation and feedforward pattern association. The event must bind its participants, location and context, so that a partial cue later reinstates the appropriate conjunction. Merely classifying inputs or remembering a location is insufficient to explain an episodic memory.

A concrete pathway begins with distributed cortical input through the entorhinal cortex. The dentate gyrus expands and sparsifies this input: strong synaptic competition means that only a small fraction of granule cells respond strongly. A winning unit or winning sparse population strengthens its active afferents through constrained Hebbian learning. Similar episodes can then recruit less-overlapping outputs, providing pattern separation before storage. Separating two nearly identical episodes is useful because a recurrent memory otherwise tends to merge their cues. The desired separation must preserve relevant conjunctions; indiscriminately randomizing the input would make later retrieval impossible.

The sparse code recruits a hippocampal CA3 ensemble. Rapid, associatively induced long-term potentiation is a candidate substrate for storing the conjunction, with its induction conditions and persistence requiring experimental constraints. Its recurrent collateral synapses allow coactive elements of the event to become associated through Hebbian learning. One idealized centred storage rule iswhere $a$ is mean coding activity and ${\xi }^{\mu }$ is an event pattern. With inhibitory control maintaining sparsity, recurrent retrieval iterates $r\leftarrow \varphi (Wr+c-\theta )$ from cue $c$. A sufficiently informative partial cue can move activity into the stored event's basin of attraction; recurrent reinstatement supplies missing components. This is pattern completion. A Hopfield network makes the principle precise in a symmetric idealization with an energy landscape, but biological CA3 connectivity need not obey the exact symmetry or update assumptions of asynchronous Hopfield energy descent. Interference, spurious states and an incorrect initial cue can all produce erroneous completion.

The hippocampal CA1 and cortical return pathways provide heteroassociative memory. During the event, CA3 activity ${\xi }^{\mu }$ is paired with a cortical or CA1 target $z^{\mu }$, giving an illustrative mapping $A={\sum }_{\mu }{z}^{\mu }({\xi }^{\mu }{)}^{\mathsf{T}}$. When CA3 completes the event, $A\hat{\xi }$ reinstates its associated output representation. Thus competitive learning distinguishes inputs, autoassociation restores a stored conjunction, and pattern association translates the retrieved conjunction into content usable elsewhere. These are separable operations even if a real circuit performs them concurrently. CA1 also receives direct entorhinal information, so it cannot be treated as a passive relay without testing the contribution of that second input.

There is mechanistically relevant evidence beyond anatomical resemblance. In adult mice, CA3-restricted NMDA receptor deletion spared ordinary acquisition and retrieval of water-maze reference memory with the full cue set, but impaired retrieval when only some cues remained. CA1 place-related activity also deteriorated under partial-cue conditions. Full-cue performance is an important control against interpreting the deficit as a general inability to swim, see cues or express a learned response. This fits the prediction that recurrent associative plasticity is particularly important for pattern completion. Nevertheless, the deletion affects receptor-dependent processing throughout the targeted CA3 cells, and does not isolate only recurrent collateral plasticity or the exact storage rule.^[ref]

A separate delayed matching-to-place experiment found impaired rapid acquisition of novel platform locations after CA3-restricted NMDA receptor deletion, with previously experienced locations relatively spared. That extends the interpretation from partial-cue recall to rapid storage of new experiences; it also cautions against assigning CA3 an exclusively retrieval-related role.^[ref]

Simultaneous dentate gyrus and hippocampal CA3 recordings offer a more direct input–output comparison. In behaving rats, conflict between local and global spatial reference frames disrupted dentate population patterns, while CA3 maintained a more coherent representation related to the stored map. Comparing the two populations under the same manipulation supports the complementary pattern separation and pattern completion computations more specifically than finding different firing rates in different areas. It still does not prove that a particular recalled human episode corresponds to one fixed CA3 attractor.^[ref]

The evidence supports the computational division of labour, but the simplest implementation is incomplete. The same region can separate strongly changed inputs and complete moderately degraded ones; separation and completion are operating regimes, not immutable labels for every trial. Fast storage must coexist with protection of old memories, and recurrent completion must not overwrite a new but similar event. Novelty, inhibition and synaptic plasticity regulation are therefore functional requirements, rather than arbitrary additions. Experimental interventions can affect acquisition, retrieval, exploration and sensory input differently, so their timing and behavioural controls matter.

An unordered fixed-point memory also omits the temporal structure of an episode. The same people in the same place at different times need contextual or sequential representations; an attractor for a bag of features does not distinguish reversed event order. Rehearsal and longer-term cortical learning may change the dependence on the hippocampus. The theory should consequently predict partial-cue success, lure errors, interference and pathway-specific deficits across time, while acknowledging that neither a spatial task nor a correlation in population activity establishes the whole of episodic memory. The combined architecture supplies an explicit storage-and-retrieval mechanism, with experimentally supported components and clear remaining requirements.

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.

A strict Lyapunov function is a continuously differentiable function $V$ with $V(0)=0$, $V(x)>0$ off the origin, and $\nabla V\cdot f<0$ off the origin in the domain. Its compact sublevel sets inside the domain are forward invariant and force convergence to the equilibrium. The domain of stability, or basin of attraction, consists of initial points whose forward solutions exist for all positive time and converge to the fixed point.

For $V=(x^2+y^2)/2$, direct differentiation and factorization giveIf $R^2=x^2+y^2<1/2$, then $x^2+4y^2\le 4R^2<2$, so $\dot{V}<0$ off zero. For any initial point in that disc, choose a slightly larger closed disc still inside it. The vector field points inward on its boundary; the trajectory is bounded, exists globally, and the strict Lyapunov decrease forces its limit to be zero. Thus the origin is asymptotically stable and the whole open disc lies in its basin.

If $R^2>2$, then $x^2+4y^2\ge R^2>2$, so $\dot{V}>0$. A solution starting there cannot cross inward through $R^2=2$ and cannot approach zero. Therefore

Put $g(z)=s(az)=m{a}^2{z}^2/({\sigma }^2+a^2{z}^2)$. The nullclines areAny steady state has nonnegative coordinates because $g\ge 0$. For $a\ne 0$, $g$ is strictly increasing on the positive half-line. If a steady state had $x>y$, then $x=g(y)<g(x)=y$, a contradiction. The case $y>x$ is identical, so all steady states lie on the diagonal. Write $x=y=z$. Besides $z=0$, the equation $z=g(z)$ givesand henceThus three distinct steady states exist exactly when $|a|m>2\sigma$, with $a\ne 0$ and the positive parameter assumptions stated above. For positive excitatory coupling this is $am>2\sigma$. At equality the two positive states merge; below it only the origin remains. If $a=0$, the origin is again the only steady state.

Both nullclines begin at the origin and approach the limiting activity $m$ along their saturating direction; they are reflections of each other across $x=y$. Flow points to the right where $x<g(y)$ and to the left where $x>g(y)$. It points upwards where $y<g(x)$ and downwards where $y>g(x)$. The plot shows these nullclines and typical trajectories for the parameters of the next part.

The origin and upper diagonal state attract nearby trajectories, while the lower positive state is a saddle. Its stable manifold separates the two attracting regions: trajectories in each basin of attraction approach its stable node, giving bistability. The diagonal is invariant, and on it the scalar activity increases between $z_{-}$ and $z_{+}$ and decreases between zero and $z_{-}$ or above $z_{+}$. Activities remain nonnegative and bounded because $0\le g\le m$. The vector field has constant negative divergence $-2/\tau$, so the Bendixson-Dulac criterion also rules out periodic orbits in the physical quadrant. The network is a switch between persistent activity levels, rather than an oscillator.

For general $k$, the same energy gives $\dot{E}=-k{y}^2$. For $k=0.1$ the eigenvalues at the origin are $-0.05\pm i\sqrt{3.99}/2$, so it is a stable focus. At $(1,0)$ they are $(-0.1\pm \sqrt{4.01})/2$, so it remains a saddle equilibrium. The vector field has constant divergence $-0.1$: there are no periodic orbits, by the planar divergence criterion. Nor can the former homoclinic orbit survive, since $E$ strictly decreases along every nonconstant orbit except at isolated crossings of $y=0$.

The stable manifold of the saddle separates escape to positive $x$ from convergence to the origin. Its two branches approach the saddle in forward time with tangent slope $(-0.1-\sqrt{4.01})/2$. The left unstable branch spirals into the origin, whereas the right unstable branch escapes. The shaded component on the origin's side of the stable manifold in the right panel is the basin of attraction. The boundary itself is excluded: its trajectories converge to the saddle. This boundary is the damped stable manifold, rather than the conservative energy contour. energy decrease prevents bounded recurrent motion, and every bounded orbit off the saddle's stable manifold consequently converges to the origin. The plotted manifold was obtained by integrating backwards from the saddle's stable eigendirection.

The only equilibrium point is $(0,0)$. Its Jacobian matrix is $\left(\begin{array}{cc}-\mu & 1\\ -1& 0\end{array}\right)$, with characteristic polynomial ${\lambda }^2+\mu \lambda +1$. For $\mu >0$ both eigenvalues have negative real parts, so the origin is locally asymptotically stable.

Use the Lyapunov function $V=(x^2+y^2)/2$. Along a solution,A Lyapunov function is positive away from the equilibrium and nonincreasing along trajectories; its sufficiently small sublevel sets prove stability. Asymptotic stability additionally means nearby trajectories converge to the equilibrium. The basin of attraction is the set of initial states whose forward solutions exist for all positive time and converge to it. Every closed disc of radius at most $\sqrt{3}$ is positively invariant, since $\dot{V}\le 0$ there. The LaSalle invariance principle says that a solution in a compact positively invariant set where $\dot{V}\le 0$ approaches the largest invariant subset of $\{\dot{V}=0\}$. Here that subset is only the origin: on $x=0$, remaining on the line would require $y=0$; the isolated points $x=\pm \sqrt{3},y=0$ on the outer disc do not remain there because $\dot{y}=-x\ne 0$. Hence the closed disc $x^2+y^2\le 3$ lies in the basin of attraction.

This estimate does not assert that the basin is exactly that disc, or the whole plane. For example, choose $R$ so large that $R^2\ge 6(\mu +1)/\mu$ and $R^2>3(\mu +2)/\mu$. The region $x\ge R,y\ge -x$ is positively invariant: on $x=R$ one has $\dot{x}>0$, and on $y=-x$ one has $(x+y{)}^{\cdot }=\mu x^3/3-(\mu +2)x>0$. Within it $\dot{x}\ge \mu x^3/6$, which forces finite-time escape. Thus the attraction is genuinely local. The Lyapunov function gives a rigorous sufficient domain for attraction without identifying its entire boundary.

For $k\ne 0$, put $t=k^2>0$ and use the positive-definite Lyapunov function $V=x^2+t{y}^2$. Its derivative isFor $V>0$ set $u=x^2/V\in [0,1]$. The brace divided by $V^2$ is $A{u}^2+2Bu(1-u)+C(1-u{)}^2$, where $A=1$, $B=\beta (1+t)/(2t)$ and $C=1/t$. Since $\beta >2$, $B>A,C$. Completing the square gives its maximumHence $\dot{V}\le -2V+2M{V}^2$. If $V(0)<1/M$, this sublevel region is positively invariant: $V$ decreases, and $\dot{V}\le -2(1-MV(0))V$. The exponential bound makes $V\to 0$. The trajectory is bounded in a compact ellipse, so smoothness of the vector field also guarantees existence for every positive time. ThereforeFor $k=0$ the displayed strict inequality has no solutions; negative $k$ gives the same ellipse as $-k$. Taking the union of the nonempty ellipses is valid because each point needs only one Lyapunov certificate.

At $k=1$ the certified disk has squared radius $2/(\beta +1)$. Along $y=0$, the limiting threshold as $k^2\to \infty$ is $4(\beta -1)/{\beta }^2$, which is strictly larger, since their difference is $2({\beta }^2-2)/({\beta }^2(\beta +1))>0$ for $\beta >2$. Choose $x^2$ strictly between these thresholds and then a sufficiently large finite $k$. This point is certified by that ellipse but lies outside the $k=1$ disk. Thus the union gives a strictly larger certified region. It need not equal the whole basin of attraction.

Let ${\varphi }_t(x_0)$ denote the flow map generated by $\dot{x}=f(x)$.

is nonpositive on $E$.

The First Lyapunov theorem says that the existence of such a function makes the origin Lyapunov stable. To prove it, choose a sufficiently small closed ball ${\overline{B}}_{\epsilon }\subset E$. By compactness and positive definiteness,By continuity, some $\delta >0$ satisfies $V(x)<\alpha$ whenever $\Vert x\Vert <\delta$. Along a trajectory beginning in $B_{\delta }$, $V$ cannot increase. Such a trajectory therefore cannot reach the sphere $\Vert x\Vert =\epsilon$, where $V\ge \alpha$. Hence it remains in $B_{\epsilon }$ for every $t\ge 0$, which is Lyapunov stability.

The LaSalle invariance principle says that if a trajectory remains in a compact positively invariant set on which $\dot{V}\le 0$, then it approaches the largest invariant subset of $\{x:\dot{V}(x)=0\}$.

For $0<k\ll 1$, the same three fixed points remain. At the origin the Jacobian isIts eigenvalues have negative real part; for small $k$ they are complex, so the origin is a stable spiral. At $x=\pm 1$, $U^{\prime \prime }(\pm 1)<0$, making the Jacobian determinant negative, so both points remain saddles.

Every nonstationary trajectory moves across the former energy contours in the direction of decreasingThe unstable branches of the two saddles directed into the central region spiral into the origin, while their outward branches escape toward $x=\pm \infty$. The stable manifolds of $(\pm 1,0)$ are the separatrices that form the boundary of the basin of attraction of the origin. Initial points between these stable separatrices lose enough energy to remain between the barriers and spiral to the origin; points outside escape to one of the two infinities.

This gives the weakly damped damped rational double-barrier phase portrait and identifies the requested domain of stability as the open region bounded by those stable manifolds.

For $k=1$, the interval in part (b) contains one exactly when $a>1$. With $V=x^2+y^2=r^2$,because $2xy\le r^2$. Hence $\dot{V}<0$ throughoutEvery trajectory starting in the disc $x^2+y^2<a-1$ remains there and approaches the origin. Thus this disc is contained in the basin of attraction of the origin.

With $s=0$, the equilibria satisfyBesides $g=0$, the two positive equilibria arewhere $g_{-}{g}_{+}=1$. Sincethe nonzero equilibrium relation $k=(1+g^2)/g$ givesThe fixed point stability for an autonomous differential equation therefore shows that $g=0$ and $g_{+}$ are stable, while $g_{-}$ is unstable. The graph of $f(g,0)$ starts at zero with negative slope, crosses upward at $g_{-}$, crosses downward at $g_{+}$, and tends to $-\infty$ as $g\to \infty$.

For a constant input $s$, write the equilibrium equation asThe low stable equilibrium and the intervening unstable equilibrium coalesce in a saddle-node bifurcation. More precisely, let $g_c$ be the first positive solution ofand defineEquivalently, $f(g_c,s_c)=0$ and ${\partial }_{g}f(g_c,s_c)=0$, with $g_c$ on the low-concentration branch. If $s_1>s_c(k)$, that branch no longer exists. Holding the input long enough carries the trajectory into the basin of attraction of the high state. When the input returns to zero, the concentration converges toThis is the saturating autocatalytic switch.

For $k\gg 1$, the threshold occurs at $g=O(k^{-1})$, soThe approximate double-root conditions areThus $g_c\simeq 1/(2k)$ andHence the constant in $s_c(k)\simeq C{k}^{-1}$ is $\boxed{C=1/4}$, as recorded by the strong-autocatalysis switching threshold.

With three distinct positive roots of an equation, the negative-leading cubic polynomial has the sign patternEquivalently its derivative is negative at the first root of an equation, positive at the middle root of an equation, and negative at the final root of an equation. The phase line arrows therefore point toward the two outer equilibrium points and away from the middle one:The middle equilibrium point separates the two basin of attraction in the bistability mathematical model parameter interval.

Reinstatement of a stored representation from an incomplete or degraded cue. Recurrent autoassociative memory provides one mechanism; its success depends on the cue entering the correct basin of attraction. Partial-cue deficits after CA3-targeted NMDA receptor deletion support a hippocampal role without identifying a unique weight-update rule.^[ref]