Emergence of local irreversibility in complex interacting systems

Living systems are fundamentally irreversible, breaking detailed balance and establishing an arrow of time. But how does the evident arrow of time for a whole system arise from the interactions among its multiple elements? We show that the local evidence for the arrow of time, which is the entropy production for thermodynamic systems, can be decomposed. First, it can be split into two components: an independent term reflecting the dynamics of individual elements and an interaction term driven by the dependencies among elements. Adapting tools from nonequilibrium physics, we further decompose the interaction term into contributions from pairs of elements, triplets, and higher-order terms. We illustrate our methods on models of cellular sensing and logical computations, as well as on patterns of neural activity in the retina as it responds to visual inputs. We find that neural activity can define the arrow of time even when the visual inputs do not, and that the dominant contribution to this breaking of detailed balance comes from interactions among pairs of neurons.

Figure 1

Irreversibility and multipartite dynamics. (a, b) A simple three-state system, with states $x$ represented as circles and joint transition probabilities $P(x\to x^{\prime })$ as arrows. (a) In a reversible system, there are no net fluxes of transitions between states, the dynamics obey detailed balance, and there is no evidence for the arrow of time. (b) Irreversible systems exhibit net fluxes between states, thereby breaking detailed balance and establishing an arrow of time. (c) A multipartite system composed of two binary spins. Only one spin is allowed to change state at a time, thus disallowing the transitions indicated by red arrows.

Figure 2

Independent and interaction irreversibility in a sensing system. (a) Sensing system with an environmental variable $x$ (left) and a sensing variable $y$ (right), each with three states. At each point in time, one of the two variables is updated at random. With probability $p_x$, the environmental variable $x$ rotates clockwise, and with probability $p_y$, the sensing variable $y$ copies $x$. (b) Fluxes between states of the sensing system (for $p_x,p_y>1/3$) induced by the environmental variable (blue) or the sensing variable (red). (c, d) Irreversibility $\dot{I}$ (c), independent irreversibility ${\dot{I}}^{\text{ind}}$ (d), and interaction irreversibility ${\dot{I}}^{\text{int}}$ (e) as functions of $p_x$ and $p_y$. While $\dot{I}$ grows with both $p_x$ and $p_y$, ${\dot{I}}^{\text{ind}}$ only increases with $p_x$, and ${\dot{I}}^{\text{int}}$ mostly increases with $p_y$, thereby distinguishing the two sources of irreversibility in the system.

Figure 3

Decomposing irreversibility in noisy logical functions. (a) System of three binary variables $x$, $y$, and $z$, where $z$ performs a noisy logical function on the inputs $x$ and $y$. At each point in time, one of the variables is updated at random. With probability $p_{\text{flip}}$, the inputs $x$ and $y$ change value, and with probability $p_{\text{error}}$, the output $z$ fails to perform the specified function (see Appendix pp5). (b, c) Fluxes between states of the system when $z$ either copies $x$ (b) or copies $y$ (c). (d) Irreversibilities $\dot{I}$ (full), ${\dot{I}}_{\text{int}}^{\left(2\right)}$ (pairwise interaction), and ${\dot{I}}_{\text{int}}^{\left(3\right)}$ (triplet interaction) vs $p_{\text{error}}$ for different values of $p_{\text{flip}}$. For both of the copy functions, irreversibility arises entirely from pairwise dynamics ($\dot{I}={\dot{I}}_{\text{int}}^{\left(2\right)}$), while triplet dynamics do not contribute (${\dot{I}}_{\text{int}}^{\left(3\right)}=0$).

Figure 4

Decomposing irreversibility in and, or, and xor functions. (a, b) For logical systems defined as in Fig. 3, we illustrate the fluxes between states when $z$ executes either and (a) or or (b). (c) Irreversibilities $\dot{I}$ (full), ${\dot{I}}_{\text{int}}^{\left(2\right)}$ (pairwise interaction), and ${\dot{I}}_{\text{int}}^{\left(3\right)}$ (triplet interaction) of the and and or systems vs $p_{\text{error}}$ for different values of $p_{\text{flip}}$. Irreversibility arises from a combination of pairwise (${\dot{I}}_{\text{int}}^{\left(2\right)}>0$) and triplet (${\dot{I}}_{\text{int}}^{\left(3\right)}>0$) dynamics. (d, e) Fluxes (d) and irreversibilities (e) when $z$ performs xor. Irreversibility arises entirely from triplet dynamics ($\dot{I}={\dot{I}}_{\text{int}}^{\left(3\right)}$), while pairwise dynamics are reversible (${\dot{I}}_{\text{int}}^{\left(2\right)}=0$). (f, g) Minimum irreversibilities ${\dot{I}}^{\left(k\right)}$ (f) and interaction irreversibilities ${\dot{I}}_{\text{int}}^{\left(k\right)}$ (g), normalized by the full irreversibility $\dot{I}$, as functions of the order $k$ for different logical functions. The errorbars for and and or reflect the small variability in ${\dot{I}}^{\left(k\right)}/\dot{I}$ and ${\dot{I}}_{\text{int}}^{\left(k\right)}/\dot{I}$ over the range of different $p_{\text{error}}$ and $p_{\text{flip}}$ values.

Figure 5

Broken detailed balance in a group of neurons. (a) Dots mark the times of action potentials (“spikes”) from 53 neurons in the salamander retina responding to a visual stimulus (see Appendix pp7 for experimental details). (b, c) The same 53 neurons are exposed to three different stimuli: a natural movie of fish swimming (b), a horizontal bar whose movement is defined by a Brownian particle on a spring (c), and the same Brownian bar in panel (c) but with one trajectory repeated multiple times. (d) To record multipartite transitions [see Fig. 1], we slide a window of width $\mathrm{\Delta }t=20\mathrm{ms}$ along the time series. A neuron transitions to active when a spike enters the front of the window (left), or inactive when a spike exits the back of the window (center). Self-transitions can occur when a cell spikes twice within the same window (right). (e–h) A random group of three neurons responding to a natural movie. (e) Probabilities $P(x\to x^{\prime })$, where black entries indicate transitions that are disallowed under multipartite dynamics. (f) Changes in state probabilities $\mathrm{\Delta }P\left(x\right)$ are small relative to their standard deviations ${\sigma }_{\mathrm{\Delta }P\left(x\right)}$, indicating that the system is in steady state. (g, h) Fluxes $P(x\to x^{\prime })-P(x^{\prime }\to x)$ illustrated as a matrix (g) and as a directed network (h). The presence of fluxes demonstrates that the neurons break detailed balance, defining an arrow of time.

Figure 6

Stimulus dependence of local irreversibility. (a) Distributions of local irreversibilities $\dot{I}$ for five-cell groups responding to a natural movie (blue), a Brownian bar (red), and a repeated Brownian bar (green). (b) The same as panel (a), but normalized to bits per second to account for variations in spike rates across stimuli. In panels (a, b), out of 100 random five-cell groups, we include only those with significant local irreversibility (see Appendix pp8). (c, d) Local irreversibility (c) and after normalizing to bits per second (d) for different stimuli as functions of the number of cells in a group. Data points are averaged over 100 random groups.

Figure 7

Decomposing local irreversibility in neuronal activity. (a) Distributions of interaction irreversibilities ${\dot{I}}_{\text{int}}^{\left(k\right)}$ of different orders $k$ for five-cell groups responding to a natural movie. (b, c) Minimum irreversibilities ${\dot{I}}^{\left(k\right)}$ (b) and interaction irreversibilities ${\dot{I}}_{\text{int}}^{\left(k\right)}$ (c), normalized by the true local irreversibilities $\dot{I}$, as functions of the order $k$ averaged over five-cell groups. (d) The fraction of pairwise irreversibility ${\dot{I}}_{\text{int}}^{\left(2\right)}/\dot{I}$ increases significantly with the local irreversibility $\dot{I}$ for five-cell groups (Spearman coefficient $r=0.31$, $p<{10}^{-3}$). In all panels, out of 100 random groups, we include only those with significant local irreversibility for each stimulus.

Figure 8

Correcting for finite-data effects on local irreversibility. (a) Estimated local irreversibility $\dot{I}$ (black markers) vs inverse data fraction for one group of $N=5$ neurons responding to a natural movie. Gray line indicates a linear fit, and red marker indicates the extrapolation to infinite data. (b) Estimated local irreversibility (black markers), linear fits (gray lines), and extrapolation to infinite data (red marker) after repeating the process in panel (a) 100 times for the same group of neurons. Data points and error bars reflect averages and standard deviations over the 100 repetitions.

Figure 9

Estimated local irreversibilities of real and time-shuffled data. (a, b) Distributions of local irreversibility estimates $\dot{I}$, normalized by standard deviations ${\sigma }_{\dot{I}}$, for real time series (a) and after randomizing spike times (b). Distributions are over the same 100 groups of $N=5$ cells analyzed in Figs. 6 and 7, and the dashed line indicates the threshold for significance. (c, d) Fraction of cell groups with significant local irreversibility as a function of group size $N$ for real time series (c) and after randomizing spike times (d). For each size, fractions are computed over 100 random groups.

Figure 10

Neurons operate at stochastic steady states. (a–c) Distributions of changes in state probabilities $\mathrm{\Delta }P\left(x\right)={\sum }_{x^{\prime }}P(x^{\prime }\to x)-P(x\to x^{\prime })$, normalized by the standard deviation ${\sigma }_{\mathrm{\Delta }P\left(x\right)}$, for groups of $N=5$ cells responding to a natural movie (a), moving bar stimulus (b), and a repeated moving bar stimulus (c). Distributions are over the $2^N=32$ different states for the 100 random groups analyzed in Figs. 6 and 7.