Self-organization, Memory and Learning: From Driven Disordered Systems to Living Matter

TL;DR

This work surveys how driven disordered systems, notably amorphous solids under athermal quasistatic driving, spontaneously organize memory-bearing states through a persistent network of soft spots. Memory and reversibility are analyzed via a transition graph ($t$-graph) topology, where elastic branches and hysteretic transitions form structures such as loop RPM and strongly connected components, with tools like the Preisach model clarifying history encoding. The authors connect these nonliving mechanisms to living systems by highlighting dimensional reduction, soft modes, and internal representations that enable anticipatory adaptation, including parallels to bacterial chemotaxis and the lac operon. They further explore random and active driving as routes to memory formation, the mesoscale QMEP model for amorphous solids, and the broader implications for pattern recognition and learning in biology, offering a unifying perspective on how simple organisms might leverage driven self-organization for environmental sensing and adaptation.

Abstract

Disordered systems subject to a fluctuating environment can self-organize into a complex history-dependent response, retaining a memory of the driving. In sheared amorphous solids, self-organization is established by the emergence of a persistent system of mechanical instabilities that can repeatedly be triggered by the driving, leading to a state of high mechanical reversibility. As a result of self-organization, the response of the system becomes correlated with the dynamics of its environment, which can be viewed as a sensing mechanism of the system's environment. Such phenomena emerge across a wide variety of soft matter systems, suggesting that they are generic and hence may depend very little on the underlying specifics. We review self-organization in driven amorphous solids, concluding with a discussion of what self-organization in driven disordered systems can teach us about how simple organisms sense and adapt to their changing environments.

Figures (5)

• Figure 1: The phenomenology of yielding and memory formation in an amorphous solid under athermal quasistatic (AQS) shear.(a) Binary mixture of particles in a 2d box undergoing shear deformation at strain $\varepsilon$. (b) Stress-strain response of the system in (a) under uniform increasing strain. The build-up of stress $\Sigma$ is punctuated by stress-drops due to plastic events, continuing until the system yields and $\Sigma$ fluctuates around $\Sigma_y$, the yield stress. (c) Plastic events are caused by spatially localized rearrangements of particles (shaded red area), called shear transformations or soft-spots. (d) Steady-state behavior of the configurational potential energy due to application of cyclic shear at amplitude $\varepsilon = 0.07$, resulting in a microscopically periodic response with repeatedly and reversibly triggered plastic events. (e) Illustration of a training protocol of $\mathcal{N}$ shearing cycles applied at amplitude $\varepsilon_T$, leading to the trained state $T$. This is followed by a read-out where a single shear cycle at amplitude $\varepsilon_R$ is applied to copies of $T$, and a distance $d(T,R)$ between the particle configurations at $T$ and $R$ is computed. (f) Evolution of $d(T,R)$ as a function of $\varepsilon_R$ for various durations $\mathcal{N}$ of training. The kink around $\varepsilon_T \approx \varepsilon_R$ develops already after a few training cycles. (g) Experimentally obtained read-out curve obtained upon training a system of polysterene particles confined to an oil-water interface. (h) Read-out response from training a mesoscale model of an amorphous solid at various amplitudes $\varepsilon_T$, as color coded in darkening shades of blue and indicated by dashed verticals. Panels (b) -- (d) adapted or plotted from data obtained in mungan2019networks, (f) adapted from fiocco2014encoding, (g) adapted from keim2022ringdown (CC BY 4.0), (h) adapted from kumar2024self.
• Figure 2: Transition graph-description of the AQS response of an amorphous solid subject to shear.(a) Viewed in the configuration space of particles, the AQS response to external shear $\varepsilon$ leads at first to continuous deformations of particle configurations $\bm{x}$, tracing out a hypercurve segment in configuration space, called an elastic branch. Its termination points mark the onset of mechanical instabilities which lead to other elastic branches, e.g. transitions from elastic branch $A$ to $B$ and $C$. Here and in the following we will denote elastic branches by capital italic letters while we will reserve the bold face letters $\mathbf{U}$ and $\mathbf{D}$, in black and red respectively, to indicate the transition from the upper and lower stability limit of an elastic branch. (b) The transitions can be represented as a directed graph whose vertices are the elastic branches. (c) Transition graphs ($t$-graphs) can be sampled from standard molecular dynamics simulations and reveal complex topological features. Each deformation protocol corresponds to a path on the graph. The evolution towards periodic response under cyclic shear applied to the glass $O$ at various amplitudes are shown by the dashed pathways. (d) The limit cycle reached from $O$ with training amplitude $\varepsilon_T = 0.05$. Top left: the training by oscillatory shear leads to the periodic response $T \to Y \to X \to T$. Center: Detailed view of $t$-graph associated with the $0.05$ limit-cycle marked in (c). Transitions under increasing and decreasing strain are shown as black/gray and red/orange arrows, respectively. The periodic response corresponds to the graph cycle traced out by starting from the vertex $X$, following the black arrows until $Y$, and subsequently following the red arrows back to $X$. The graph-cycles are labeled by their extreme points, e.g. $(X,Y)$, and exhibit hierarchical nesting, e.g. $(F,L)$ within $(F,Y')$ within $(X,Y')$ within $(X,Y)$. Nesting is characteristic of loop return-point-memory munganterzi2018. (e) The transitions forming the main hysteresis cycle $(X,Y)$ are due to $13$ soft spots whose sample locations are indicated by green ellipses. The labels on some transitions shown in (d) indicate the soft-spots involved. Panels (c) -- (e) adapted from mungan2019networks.
• Figure 3: Mesoscale emergence of "soft spots" in a sheared granular packing. (a) Sketch (top view) of a shear cell containing glass beads vertically confined by a transparent load. Shear can be applied using a central vane (in red) to which a controlled torque is applied and the angular rotation measured. (b) Shear stress evolution at constant external stress rate (stress ramp) showing a sequence of sudden stress drops. (c) Direct visualization, during the charging process of "hot spots", i.e. loci of large deformation spanning about 5-10 grain sizes, using an optical Diffusing Wave Spectroscopy (DWS) technique. The darker the color, the larger the deformation. The stress drop at $\sigma_r$ corresponds to the spatial accumulation of "soft spots" around a large shear band zone. (d) Under constant shear stress, the logarithmic creep (in blue) quantitatively corresponds to the cumulated value $N(t)$ (in red) of all visualized "soft spot" events, thus revealing the material support of an internal slowly varying variable interpreted as the "fluidity" parameter in soft glassy rheology models. (e) Under external stress modulation around a constant value, the logarithmic creep becomes linear due to the long-term accumulation of irreversible events materialized by a constant production rate of "soft spots", corresponding equivalently to a dynamical irreversibility of the fluidity variable. Figure adapted from refs Amon2012 and Pons2015.
• Figure 4: The Preisach model.(a) A Preisach hysteron $i$ as a two-state system with hysteretic transitions between states 0 and 1 at switching fields $\varepsilon^\pm_i$. (b) The two possible main hysteresis loop-graphs of a system of $N = 2$ independent hysterons. Such graphs are called Preisach graphs. Vertices are labeled by hysteron configurations. Labels next to transitions indicate the hysteron that changes state. Hysterons switch states independently and we label them according to the order in which they switch from state 0 to 1, see black labels next to transitions from 00 to 11. This leaves the order in which hysterons switch back to 0 undetermined, which we prescribe by $\rho$. For $N = 2$ hysterons, the possible switch-back sequences are $\rho = 1\,2$ and $\rho = 2\,1$, as indicated by the red labels next to the transitions from 11 to 00. (c) The Preisach graphs of $\rho = 2\,3\,1$ and $\rho = 2\,3\,4\,1$. Due to loop return-point memory, the Preisach graph of $\rho = 2341$ can be described as a merger of the $\rho = 2\,3\,1$ graph (inside orange rectangle) and a certain subgraph (green rectangle inside) that is copied and connected to it by the two transitions involving the state changes of the fourth hysteron. (d) The number of vertices of the Preisach graph of $\rho$ is equal to number of increasing subsequences contained in $\rho$. Each increasing subsequence can be seen as a deformation history that is applied to ${\tt 00\ldots 0}$, mapping it thereby to a distinct vertex of the Preisach graph. This is illustrated for the case $\rho = 2\,1\,8\,4\,5\,3\,6\,7$ and the increasing subsequence $2, 4, 5, 6, 7$, whose deformation path leading to $\bm{\sigma}$ is shown by the $\mathbf{U}$- and $\mathbf{D}$-transitions higlighted in black and red, respectively.
• Figure 5: Self-organization and memory in a mesoscale model of an amorphous solid subject to random shear loading (panels ordered column-wise).(a) (Left) 2d lattice of mesoscale cells. (Right) The response of each cell $(i,j)$ to an external stress $\Sigma$ follows a sequence of local elastic branches, labeled $\ldots, \ell-1, \ell,\ell + 1, \ldots$, which are each limited by stress thresholds in the forward $\sigma_{ij,\ell}^+$ and reverse $\sigma_{ij,\ell}^-$ directions of shear. A mesostate $A$ corresponds to the matrix $\ell_{ij}$, indicating the local branch index of each cell. (b) Sample history by application of random strains $\varepsilon$ to an initial glass $O_0$. The strain protocol is a random walk along the strain axis with fixed steps $\delta \varepsilon$ and reflecting boundaries at $\pm \varepsilon_{\rm max}$. The boundary reached first defines the sense of driving and is called the first boundary. A cycle is defined in terms of the first passage times to go from zero strain to the first boundary, then to the opposite boundary and to subsequently return to zero strain, like the cycle $O_0 \to Y_1 \to X_1 \to O_1$ shown. (c) Definition of the (deterministic) 1- and 2-cycle read-out protocols. In the latter, a second read-out cycle of the same amplitude is applied to $T$. The configurations reached at the end of the first and second read-out cycle are $R$ and $R_2$. (d) Results of 1-cycle read-out applied to the trained state $T$ reached after various numbers of random training cycles, cf. Fig. \ref{['fig:amorphous-intro']}(f). (e) 1-cycle read-outs from glasses subjected to 20 random cycles at different amplitudes $\varepsilon_{\rm max}$, as indicated by dashed verticals. (f) 2-cycle read-out of $d(R,R_2)$, cf. (c). Panels (a) -- (f) adapted from kumar2024selfmungan2024self.