Econophysics
Here I discuss some potential applications of my work on neural networks to the field of econophysics. The topics considered in the following sections are:
1) Critical Slowing Down and Early-Warning Signals;
2) Imitative Behavior in the Ising Model and Extreme Value Theory;
3) Strategic Decisions, Mean Field Theory and Graphics Processing Units (GPUs).
Early-Warning Signals
Introduction
In [1] (see also [2] for further results) we focused on the study of bifurcations in stochastic neural networks. In dynamical systems theory, the term “bifurcation” refers to a sudden, qualitative change of the system dynamics, which occurs when varying gradually one or more model parameters. Specifically, in [1] we studied the changes of stochastic dynamics that occur near 3 kinds of bifurcations, when varying the external input currents to the network. Those bifurcations are known as “Limit Points” (LP, aka “tipping points”), “Hopf” (H), and “Branching Points” (BP). When the network undergoes those bifurcations, we observe, respectively, a sudden and strong shift of the (mean) neural activity, the formation/annihilation of neural oscillations, or variations in the degree of heterogeneity of the neural activity (through a phenomenon known in physics as “spontaneous symmetry breaking”).
Moreover, the network dynamics near all these bifurcations becomes dominated by a phenomenon known as “critical slowing down” (CSD). This is where system dynamics becomes progressively less resilient to perturbations as the bifurcation point approaches, causing dynamics to become more variable and auto/cross-correlated.
Interestingly, for a given kind of bifurcation, CSD occurs independently of the specific mechanisms involved in bringing about that bifurcation. For this reason, CSD is an active topic of research in many scientific fields, such as biology, neurology, climate change, human physiology and economics. The following picture shows an example of CSD in marine ecosystems near a tipping point, where a small variation in an environmental parameter (stressor) can cause a rapid change from one ecological condition (coral dominated system) to another (algae dominated system).
Applications to Econophysics
The increase in the variability and correlation of the system dynamics observed during CSD can be used as early-warning signals to predict qualitative changes in a dynamical system before they occur. However, in the scientific literature, CSD is typically studied in models driven by independent sources of noise, and near LP bifurcations, in that they represent mathematical simplifications describing systems at the brink of collapse. In [1] we extended those results by studying CSD in multi-population networks with arbitrarily-correlated sources of noise, and we showed that CSD exhibits strong qualitative differences near the LP, H and BP bifurcations, in terms of variability and correlations.
In the field of economics, the technique that we applied in [1] can be adapted to shed new light on specific aspects of economic growth variability in historical GDP data, as well as to define new early-warning signals that may help to anticipate sudden shifts of dynamics (not necessarily collapses) in financial markets.
Imitative Behavior
Introduction
In [3] we applied Extreme Value Theory (EVT) to study how the dynamics of binary network models with random synaptic weights and/or topology is affected by their network parameters (note that in [3] we focused on the external input currents to the network, but this approach can be easily extended to any other network parameter). This is a powerful way to study the effect of heterogeneities of the interactions that have been observed experimentally in biological neural networks.
EVT is concerned with the study of the asymptotic distribution of extreme events, that is to say events which are rare in frequency and huge in magnitude with respect to the majority of observations. For example, in the scientific literature EVT has been used to study the occurrence of strong earthquakes, such as the 1960 Valdivia earthquake in Chile, which was the most powerful earthquake ever recorded (magnitude 9.4 – 9.6 on the moment scale).
Applications to Econophysics
Since EVT is the theory of modelling and measuring events which occur with very small probability, this implies its usefulness in financial risk modelling, as risky events per definition happen with low probability.
EVT has also been applied to study the occurrence of severe economic downturns, such as the Great Depression of 1929-1939.
In [3] we focused on the study of “bifurcations” in binary neural networks, namely of sudden, qualitative changes of the network dynamics, which occur when varying gradually the input currents (e.g., the network may have, say, three metastable states, and then switch to an astable state when the input is increased). Then we showed that bifurcations in binary neural networks are, indeed, extreme events, so that their probability to occur can be quantified by EVT.
Note that while in [3] we considered networks of binary neurons (i.e., having output 0 = not-firing, 1 = firing), the same mathematical approach can be used to study networks of Ising-like units, which have been applied extensively to describe the local imitative (Herd) behavior between agents in financial markets (namely, the Ising units have output +1 = buy, -1 = sell). In other words, when trading financial assets (i.e. when deciding whether to sell or buy), people are influenced by their “first neighbors”, and this mechanism is captured by the Ising network.
Note that in finance the Ising model is typically studied by means of equilibrium statistical physics in the thermodynamic limit (i.e. a market with infinitely many traders). Interestingly, when the “economic temperature” (i.e. the average amount of money per agent) is low, the local interaction between traders leads to a global phenomenon called “phase transition”, i.e. a sudden price jump where most of the investors polarize to a buy or to a sell state. On the other hand, the technique applied in [3] relies on dynamical systems theory. This allows one to study bifurcations in markets of any size (not necessarily infinite), and for any distribution of the links strength between traders, if the links are statistically independent.
Strategic Decisions
Introduction
In [3 – 6], we discussed mathematical and computational techniques for studying stochastic neural networks composed of a large number of neurons. In [3, 4] we considered multi-population network models of firing-rate neurons evolving in discrete time, while in [5, 6] we focused on spiking models in continuous time. Specifically, in [5, 6] we discussed how to calculate the probability distribution of the neural activity by means of Monte Carlo simulations, as well as through the numerical resolution of the mean-field (Fokker-Planck) equation of the network. Both the approaches are computationally expensive, however we showed that they can be implemented efficiently on graphics processing units, or GPUs (see also this link for further information).
Unlike central processing units, or CPUs, which perform computations sequentially by running a single thread very quickly, GPUs have a parallel architecture that runs many concurrent threads slowly.
Note that while CPUs are struggling to find ways to improve their speed, the computational power of GPUs increases dramatically every generation, as they add more functional units and processing cores. For this reason, GPUs represent an excellent platform for accelerating many scientific applications, including those in econophysics discussed below.
Applications to Econophysics
Mean-field theory and GPUs find large application also in econophysics. An important example is represented by mean-field game (MFG) theory, which studies the strategies of agents of a large population in a competitive environment. Each agent has negligible impact upon the system, and he/she seeks to maximize its own benefit, according to the actions of other agents surrounded.
Under fairly general assumptions, it can be proved that a class of MFGs is the limit of an N-player Nash equilibrium, when N goes to infinity. Moreover, in continuous time, a typical MFG consists of a Fokker-Planck equation (describing the evolution of the aggregate distribution of agents), coupled to a Hamilton-Jacobi-Bellman equation (describing the dynamics of the cost function of an individual agent). Note that the numerical methods that we developed in [5, 6] can be adapted to solve this system of equations, therefore GPUs represent an attractive source of computational power for accelerating the simulation of the agents’ strategies.
Bibliography
[3] D. Fasoli and S. Panzeri, Stationary-state statistics of a binary neural network model with quenched disorder, Entropy, 21(7):630, 2019 (URL)
[4] D. Fasoli, A. Cattani and S. Panzeri, Pattern storage, bifurcations, and groupwise correlation structure of an exactly solvable asymmetric neural network model, Neural Computation, 30(5):1258-1295, 2018 (URL)
[5] J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons, The Journal of Mathematical Neuroscience, 2(1):10, 2012 (URL)
[6] J. Baladron, D. Fasoli and O. Faugeras, Three applications of GPU computing in neuroscience, Computing in Science and Engineering, 14(3):40-47, 2012 (URL)