Information dynamics of our brains in dynamically driven disordered superconducting loop networks

TL;DR

Numerical simulations performed with a lumped circuit model of a 4-loop network show that information written through excitations is translated into stable states of trapped flux and their time evolution, revealing the universal nature of information dynamics through the stated two principles.

Abstract

Complex systems of many interacting components exhibit patterns of recurrence and emergent behaviors in their time evolution that can be understood from a new perspective of physics of information dynamics, modeled after one such system, our brains. A generic brain-like network model is derived from a system of disordered superconducting loops with Josephson junction oscillators to demonstrate these behaviors. The loops can trap multiples of fluxons that represent quantized information units in many distinct memory configurations populating a state space. The state can be updated by exciting the junctions to allow the movement of fluxons through the network as the current through them surpasses their thresholds. Numerical simulations performed with a lumped circuit model of a 4-loop network show that information written through excitations is translated into stable states of trapped flux and their time evolution. Experimental implementation on the 4-loop network shows dynamically stable flux flow in each pathway characterized by the junction firing statistics. The network separates information from multiple excitations into state categories with large energy barriers observed in simulations that correspond to different flux (information) flow patterns observed across junctions in experiments. Strong evidence for associative and time-dependent (short-to-long-term) memories distributed across the network is observed, dependent on its intrinsic and geometrical properties as described by the model. Loop network topology abstraction using the model separates the flowing patterns of information from its physical constraints and describes systems of any scale and complexity. The accuracy of flow statistics are limited by the resolution of local external measuring clock(s) revealing the universal nature of information dynamics through the stated two principles.

Figures (6)

• Figure 1: (A) Superconducting loop disordered network with multiple spiking inputs and outputs. A typical trapped flux configuration with circulating currents around individual loops is shown as $n_1\Phi_0$ and $n_2\Phi_0$ and around multiple loops shown as $n_{345}\Phi_0$. Each loop can be treated in simulations using an equivalent lumped element circuit model of inductors and junctions as shown. Feedback currents are useful to reconfigure the mapping between input signals and the resulting memory states. (B) Configuration space with disordered energy landscape with coordinates defined by all the possible trapped flux states. Stable memory configurations are given by the local energy minima. Any of the trapped flux states can be dynamically reconfigured to be local minima with suitable excitations (inputs or feedbacks). (C) Model of a 4-loop network showing possible ways (arrows) to excite the system to access various flux configurations. (D) An equivalent lumped circuit model of the 4-loop network was implemented to generate the simulation results discussed in the article. (E) YBCO-based 4-loop network with junctions defined by focused He-ion beam irradiation-induced insulating tunnel barriers labeled by white lines. (F) The number of flux configurations and the energy calculated from circulating currents from the circuit model show the density of states from zero trapped flux to maximum filled flux state for the 3-loop memory network.
• Figure 2: (A) Schematic illustrating simulations of the uniform magnetic field across the 4-loop network described in Fig. \ref{['Fig1']}(C-E). A magnetic field pulse is applied with a pulse width of $1ns$ and different pulse heights and then the system is allowed to relax to realize different meta-stable trapped flux configurations. (B) Simulation of time-evolution of the state of the system due to magnetic field excitation. The relaxed state is recorded with pico-second increments to the magnetic field pulse width to map the time evolution for $5$ different pulse amplitudes. This state is independent of time after a critical excitation time period (pulse width) dependent on the loop parameters of the involved trapped flux configurations. (C) The voltage across $5$ junctions in the 3-loop memory network of Fig. \ref{['Fig1']}(C-E) showing multiples of quantized flux pulses entering and leaving the network through each of the junctions for the magnetic field of $500\frac{\Phi}{\Phi_0}$ with a pulse width of $1 ns$. There is no flux motion after the critical excitation time period. (D) The energy of the state of the network in Fig. \ref{['Fig1']}(C-E) with the respective magnetic field for a fixed pulse width of $1 ns$ at different times during the relaxation process. Discrete states reached by $1.5 ns$ ($500 ps$ of relaxation) represent the local minima in configuration space. (E) The energy of the relaxed state is shown to demonstrate reconfiguration of the state-space with different constant feedback currents at $I_2$. Feedback current has the effect of tilting the state space to access different memory states.
• Figure 3: (A) Schematic illustrating simulations of spiking excitation induced at $J_1$ in the network described in Fig. \ref{['Fig1']}(C-E). An input flux flow at of a constant rate is applied for a fixed pulse duration of $t_1=1ns$ for different pulse heights $I_1$ and then the system is allowed to relax to realize different meta-stable trapped flux configurations. (B) The energy of the state of the network in Fig. \ref{['Fig1']}(C-E) with the respective excitation voltage $V_1$ (i.e., average flow rate of $\frac{V_1}{\Phi_0}$) for a fixed pulse width of $t_1=1ns$ observed at different time intervals (i.e., at $t_2=1.01ns$, $t_3=1.05ns$ and $t_4=1.5ns$) during the relaxation process. Discrete states reached by $1.5ns$ represent the local minima in configuration space. (C) The energy of the relaxed state is shown to demonstrate reconfiguration of the state-space with active constant feedback $I_2 = 700\mu A$ showing network behaviors such as categorization and associative memory. (D) Simulation of time-evolution of the state of the system due to local excitations (i.e., $I_1 = 1mA$ and $I_2 = 0$). The state is recorded after relaxation as a function of pulse width $t_1$ with picosecond increments to map the time evolution. (E) Firing probability $p_1$ of junction $J_6$ with respect to $J_1$ as a function of integration time $T$ during the time evolution of the state shown in Fig. \ref{['Fig3']}D. (F) Simulation of time-evolution of the state of the system due to local excitations (i.e., $I_1 = 0.9mA$ and $I_2 = 0.7mA$). The state is recorded after relaxation as a function of pulse width $t_1$ with picosecond increments to map the time evolution. (G) Firing probability $p_1$ of junction $J_6$ relative to $J_1$ shown as a function of integration time $T$ during the time evolution of the state shown in Fig. \ref{['Fig3']}F. (H) Simulation of time-evolution (i.e., the relaxed state after picosecond increments) of the state of the system due to dynamically varying input excitation, i.e., $I_1$ is linearly varied from $-1.5mA$ to $0mA$ in an interval of $6ns$ with constant feedback current $I_2 = 0.7mA$. With time-dependent input, the trapped flux configurations that generate the output flow have different temporal stability.
• Figure 4: (A) Graphical illustration of the flux flow patterns between junctions on the outer loop that are defined by the underlying flux configurations that are stable over the integration period. The flux configurations are directly related to the relative junction firing probabilities. (B) A representation of distinct circulating current (trapped flux) paths in the 4-loop network that contribute to the switching activity of the output junction $J_6$. Current paths from 1 to 4 show the contributions from the 3-loop memory while paths 5 and 6 show directly coupled paths between the input junction $J_1$ and the output junction $J_6$. Flux configuration of the network is composed of combinations of flux from all the paths. (C) Input and output flux streams can be measured as voltages $V_1$ (input), $V_2$ and $V_3$ (outputs) across $J_1$, $J_3$ and $J_6$ respectively. Output spiking probabilities are constrained by the sum rule $p_1 + p_2 \approx 1$ for long integration times in a steady state. (D) Correlation between activities of any pair of junctions can be described as due to transitions between directly coupled closed loops including both the junctions, together with transitions in closed loops encompassing either of the junctions. For the 4-loop network, the indirectly coupled loops encompassing only $J_6$ refer to the trapped flux in the 3-loop memory network shown in Fig. \ref{['Fig4']}B (1-4). All the loops are coupled to each other and $p_1$ is affected by transitions between any of these paths. (E) Network topology representation of the 4 disordered loops with nodes referring to junctions highlighting the dominant flux flow pathways. Two distinct information flow paths ($p_1$ and $p_2$) exist from input to outputs.
• Figure 5: (A) Flow strength in pathway $p_1$ in the range $-1$ to $1$ as a function of excitation currents $I_1$ and $I_2$. Transitions between flow patterns are disordered reflecting the network structure. Oscillations represent differences in temporal stabilities of states discussed in the text. (B) Flow strength in pathway $p_1$ in the range $-1$ to $1$ across the measurement state-space of $V_1$ and $V_2$ shows $8$ flow patterns and the transition edges between them. (C) Simulation results of the energy of the state with active excitations across the state space of $V_1$ and $V_2$. Energy monotonically increases with $V_1$ and $V_2$ due to the modulation from increasing excitation currents. (D) Simulation results of the relaxed state energy after excitations are turned off, plotted against the state space of $V_1$ and $V_2$ while they were active in a steady state. Saturated states in orange represent the saturated trapped flux in the 3-loop memory also shown in Fig. \ref{['Fig6']}A. Useful memory states store an output firing probability in the range $0>p_1>1$. (E) Flux flow patterns between input and output are mapped from the experimental results on the 4-loop YBCO network (Fig. \ref{['Fig1']}E) for a long integration time of $T = 1$ms. Different flow patterns characterize the memory state categories separated either by a change in flow direction along a path or large changes in relative strengths.