Hysteresis & Memory

The unifying principle

For a memoryless system, output is a pure function of input: $y(t) = f(x(t))$ . Hysteresis breaks this — the map is multivalued and path-dependent:

$y(t) = \mathcal{H}[x(\tau)]_{\tau \le t}$

The output is a functional of the entire input history. The minimal machinery to produce this is a latch with two stable states plus a threshold gap. Define an up-threshold $\alpha$ and a down-threshold $\beta$ with $\beta < \alpha$ :

y(t) = \begin{cases} +1 & \text{if } x(t) > \alpha \\ -1 & \text{if } x(t) < \beta \\ y(t^-) & \text{if } \beta \le x(t) \le \alpha \end{cases}

The middle line is the whole story: inside the dead band $[\beta,\alpha]$ the system holds its last value. This elementary unit is a hysteron. Any hysteresis loop can be built by summing many hysterons with distributed thresholds (the Preisach model):

$y(t) = \iint_{\beta \le \alpha} \mu(\alpha,\beta)\, \gamma_{\alpha\beta}[x(t)]\, d\alpha\, d\beta$

Two ingredients recur everywhere below: (1) a feedback loop with gain > 1 creating bistability, and (2) a gap between switching thresholds that stores one bit of history.

How it shows up in each field

Physics — ferromagnetic hysteresis

A ferromagnet's magnetization $M$ lags the applied field $H$ . Sweeping $H$ up and down traces the B–H loop; at $H=0$ a residual remanence $M_r$ remains — that's the stored bit of every hard drive and refrigerator magnet.

Notation: $M(H)$ multivalued; coercivity $H_c$ = field needed to flip back, i.e. the half-width of the dead band.
Why it's the same idea: neighboring spins reinforce each other's alignment (exchange coupling → positive feedback, mean-field $H_{\text{eff}} = H + \lambda M$ ). Domain walls pin on defects, giving the threshold gap.
Example: magnetize iron in a strong field, remove it, iron stays magnetized. Energy dissipated per cycle $= \oint H\, dM$ = loop area.

Hardware — the flip-flop (SR latch)

Cross-couple two NOR gates and you get a circuit whose output depends on which input last went high.

Notation: state $Q$ ; a Schmitt trigger makes the threshold gap explicit with $V_{T+} > V_{T-}$ .
Why it's the same idea: the cross-coupling is literally positive feedback (loop gain > 1 → two stable states $Q=0,1$ ); the Schmitt gap $V_{T+}-V_{T-}$ is the coercivity — noise inside it can't flip the bit.
Example: SRAM cell holds a 1 until explicitly overwritten. Hysteresis on a comparator debounces a noisy switch: the signal must fully cross $V_{T+}$ then fall below $V_{T-}$ to toggle.

Biology — neuronal bistability & memory

Neurons and gene circuits latch into states. A neuron won't fire until depolarization crosses threshold $\theta_{\text{up}}$ , and once a bistable network is "on," it takes a lower input to keep it on than to start it.

Notation: membrane potential $V(t)$ ; persistent activity in working-memory attractor networks $r_{\text{high}}$ vs $r_{\text{low}}$ .
Why it's the same idea: recurrent excitation (positive feedback among interconnected neurons) creates two stable firing rates; $\text{NMDA}$ /channel kinetics give the threshold separation. Same for the lac operon and MAPK switches in cells.
Example: prefrontal cortex "delay activity" holds a remembered digit after the stimulus is gone — the network stays in its high state (hysteresis = short-term memory). Neuronal spike-frequency adaptation and $Na^+$ channel inactivation also produce path-dependence.

Stock-Market — momentum & path-dependent expectations

Prices and trader behavior depend on the recent trajectory, not just current fundamentals.

Notation: momentum signal $m_t = P_t - \text{MA}_n(P)$ ; buy when $P$ crosses above a moving average, sell when it crosses below — with a band to avoid whipsaw.
Why it's the same idea: trend-following creates positive feedback (buying pushes price up, attracting more buyers → herding); transaction costs and the entry/exit band form the threshold gap. Prospect-theory disposition effect (holding losers, selling winners) makes decisions depend on the purchase-price reference point — literally history-dependent state.
Example: a stock in an uptrend keeps attracting momentum buyers until it decisively breaks support, then reverses — the entry and exit prices differ, tracing a behavioral hysteresis loop. Macro version: labor-market hysteresis, where a recession raises long-run unemployment permanently.

Why this bridge matters

What transfers: the Preisach/hysteron decomposition developed for magnets is used verbatim to model economic hysteresis and material fatigue. The Schmitt-trigger insight — deliberately widening the threshold gap to reject noise — is a universal design principle: neurons use it against synaptic noise, traders against price chatter, engineers against electrical noise.
Intuition unlocked: thinking of a neuron as a flip-flop clarifies working memory (you store a bit by parking a bistable loop in one state). Thinking of momentum trading as ferromagnetism explains bubbles: aligned "spins" (traders) reinforce until a field reversal cascades. Conversely, the loop area = energy dissipated from physics tells the trader that every hysteresis strategy has a built-in cost per cycle (whipsaw losses) — you pay for memory.
Design lesson: memory always costs dissipation and always needs feedback gain > 1 plus a gap. Want more stability? Widen the gap and lose responsiveness. Want faster response? Narrow it and lose noise immunity. This tradeoff is field-independent.

Connections

03 Ferromagnetism & Domain Theory
07 Preisach Model of Hysteresis
12 SR Latches & Flip-Flops
14 Schmitt Triggers & Noise Immunity
21 Bistable Attractor Networks
23 Working Memory & Persistent Activity
31 Momentum & Trend-Following Strategies
34 Behavioral Finance: Disposition Effect
40 Positive Feedback & Bistability

#bridge