Department of Biology

Quasi-stable attractor states as a framework for neural computing

When we observe an ambiguous image such as the ones below, our percept tends to flip sporadically between different interpretations of the image. For each percept the neural activity is in a quasistable attractor state. The states are attractor states because following any deviation the neural activity is "attracted" back to one of the states, just as gravity attracts a flipped coin to one of two states flat on the ground. The states are quasistable, because they do not persist indefinitely as would a stable attractor state (like a flipped coin). The transitions are to a large degree induced by noise, meaning fluctuations of neural activity that may be entirely randomly driven, akin to the ground under a flipped coin vibrating so much that the coin occasionally flips again.

In my research I seek to uncover the computational capabilities of such quasistable attractor states, and their role in decision making plus sequential memory. My approach is three-pronged:

1. Analysis of neural activity through techniques such as hidden Markov modeling, which do not require averaging of data across trials (such averaging is often necessary, but destroys the single-trial correlations and dynamics that reveal state transitions).
2. Simulations of spiking neurons in circuits designed to reproduce such state transitions. In this manner simulated neural data can be compared, via the same analyses, with the recorded neural activity.
3. Mathematical analysis of model networks to reveal the information processing capabilities of circuits groups of neurons connected randomly or otherwise.

Combining multiple forms of homeostasis

Decision-making Code

Decision-Making Code for Miller and Katz, 2011

• Simulations with 1D potential: decision_potential_1d.m
• Simulations of 2-variable model: decision_2d.m

Learning and Solving Associative Cognitive Tasks

Any thoughtful action takes into account prior knowledge and context. I am implementing computational models to explain how cortical and hippocampal networks can solve a range of tasks that require choices based on learning and context.

I am using computational models to address the following questions:

1. Why is the hippocampus necessary for solving some associative tasks, but not other, similar ones?
2. Is there a trade-off between persistent activity following an input, as needed for short-term memory, and the ability to associate one input with another?
3. Does homeostasis explain the loss of ability to solve a previously learned task upon removal of hippocampus, when the task can be solved more rapidly without the hippocampus?
4. Can associations formed in the hippocampus bias a decision-making network, enabling correct performance in a set of contextual tasks?
5. Can the same associations in the hippocampus slow learning of tasks that do not require hippocampal associations to form?

Parametric Working Memory and Sequential Discrimination

Working memory is the ability to hold information temporarily, `on-line' in preparation for its imminent use in a decision. The mechanism of this short-term memory is either through recurrent firing of neurons, particularly in the prefrontal cortex, or through persistent currents which have been measured in the entorhinal currents.

The task that I model requires monkeys to make a decision by comparing two stimuli that are separated by a delay. Ranulfo Romo and co-workers at UNAM in Mexico City train monkeys to differentiate the vibrational frequency of stimuli to their finger tips. The discrimination is sequential, with a delay between the two stimuli (hence the need for working memory). Importantly the two stimuli are to the same finger, and activate the primary sensory cortical neurons in the same manner, just at different times. A model of the cortical network hence requires a response based on whether the second stimulus is of greater or lesser frequency than the first. As such, it must perform a subtraction across time.

I am carrying out computer simulations of networks of neurons which maintain persistent activity in response to a cue stimulus. As such, they exhibit the critical features of working memory. The memory circuit is equivalent to an integrator in the mathematical sense, as a greater frequency of vibration for a fixed duration leads to greater activity in a memory store. Similarly, an integrator does not change when its input ends, so exhibits persistent activity during any delay between stimuli. Much research is ongoing to address whether neuronal integrators are essentially continuous, or more discrete, allowing for robustness.

In my model of the discrimination part of the task, I use a robust integrator to hold the memory, but assume inhibitory connections from the integrator to its inputs. This method of integral feedback control is common in engineering, and allows for a robust cancellation of the input during the first stimulus, leading to discrimination of the difference between two stimuli spaced in time.

Sequential discrimination model: computer code

Purpose
The codes simulate neurons involved in a task requiring an initial stimulus (f1), retention of the value of the inititial stimulus in memory during a delay, followed by a second stimulus (f2). The codes use integral feedback control to produce neurons tuned to the difference of the two stimuli. This allows a response to be made based on which of the two stimuli was greater.
The xppaut codes are based on firing rate models.
The C++ code simulates a network of spiking neurons.

xppaut code

• 2-unit integral feedback control with a continuous integrator.
• 2-unit integral feedback control with a discrete integrator.

C++ code

To run the code:

1. 1) Download all files.
2. Compile eg:
   g++ netinmain.cpp discrimnet.cpp structure.cpp -O3 -o dis
3. Run the executable followed by a number. Last 4 digits, N, encode the stimulus set where N=2(f1)+1,odd, if f2>f1 and N= 2(f1), even, if f1>f2. The first digits form the randum number seed.
   eg ./dis 150021 > dis.out &
   will run the code with random number seed of 15, using f1=10Hz and f2>f1, in the present version f2=18Hz. Alternatively:
   ./dis 3310044 > dis.out &
   will run the code with random number seed of 331 and uses f1=22Hz and f1>f2 such that f2=14Hz in this version.
   Adjustments: at present, the code is designed to run using the script, with f2=f1+8Hz or f2=f1-8Hz. As of now, to change the frequency difference, the code in netinmain.cpp should be altered, where the variable istrength2 is set.

• script to run C++ code: runs 14 jobs in parallel with Mosix
• C++ code with comments on how to run it: discrimnet.cpp
• C++ startup code containing `main': netinmain.cpp
• C++ code with connectivity profile setup: structure.cpp
• Header file with most parameters and their purpose: dnet.h
• Random Number generator file: MersenneTwister.h
• Input file with single neuron properties and simulation times: discrimnet_in.dat
• Input file with connection strengths: connections_in.dat

Molecular basis of long-term memory

John Lisman and coworkers have put forward a model for long-term memory based on the autophosphorylation of CaMKII holoenzymes and their dephosphorylation by PP1.Influx of calcium into the synapse can cause the process of autophosphorylation to dominate, leaving the CaMKII holoenzymes in a highly phosphorylated "up" state. The system is bistable, so that at resting calcium an unphosphorylated "down" state is stable, as well as the "up" state. Bistability occurs, because (a) autophosphorylation is cooperative within a holoenzyme and slow to get started, while (b) dephosphorylation saturates, so the per molecule rate is slower in the "up" state than "down" state.

The location of CaMKII molecules that can take part in long-term potentiation (the strengthening of a synapse believed to underly long-term memory) is the post-synaptic density. The number of CaMKII holoenzymes in the postsynaptic density is between 5 and 30, depending on its size. As in reality, all reactions occur stochastically, it is a significant question as to how stable any memory storage can be. Fluctuations will eventually cause a switch of the system between "up" and "down" states or vice versa. Any such spontaneous switching is disruptive of memory, so it is important to know the timescale of such behavior. I have carried out both detailed simulations and analytical calculations for the system of enzymes, verifying that the timescale for spontaneous transitions rises exponentially with the number of enzymes present. In a reasonable range of parameters, 20 holoenzymes are sufficient to maintain memories on the timescale of human lifetimes, even when the individual proteins are being replaced by turnover randomly on a timescale of 30 hours.

Condensed Matter Theory

Superconductivity

Superconductivity is a fascinating phenomenon, which exhibits the effects of quantum mechanics on a macroscopic scale. In particular, the electrons move coherently in a single quantum state. In conventional superconductors, this coherence has come about because of an effective attraction between electrons due to the lattice vibrations (phonons) and sees the electrons paired together (as Cooper pairs). Cooper pairs have integer spin, so have the properties of bosons, and indeed, superconductivity can be considered as the Bose-Einstein condensation of Cooper pairs.

Superconductors are named as such, because of their ability to conduct electricity with zero energy loss, or zero resistance (up to a critical current density). This is principally because the current carrying state is also a quantum ground state of the superconductor, and there is a finite energy gap (Delta) to any excited state. This means that individual electrons are scattered by impurities or thermal vibrations, which are the normal causes of resistance. At temperatures high enough such that the thermal energy (kT) is of the order of the gap (Delta) the superconducting properties disappear. Hence superconductors only work below a critical temperature, Tc, which ranges from 0-23 degrees Kelvin for conventional superconductors, but has reached over 130K for high-temperature superconductors. Note that high-temperature in this field is still at colder than -140 degrees Celsius or -220 degrees Fahrenheit!

The more fundamental signature of a superconductor, though, is its ability to expel a magnetic field from within its interior, so long as the field is below a critical value (Hc). The expulsion of magnetic field is called the Meissner Effect, and results in a repulsive force between a superconductor and any magnet placed near it. Many demonstrations of superconductors show the Meissner Effect as a small magnet levitating above the superconductor. The effect has a large scale technological application in Japan, where it is used to levitate high speed trains.

De Haas-van Alphen Oscillations in the Vortex State.

Many superconductors are of type-II and can exist over a wide range of magnetic fields, between a lower critical field (Hc-1) where magnetic flux first enters the superconductor and an upper critical field (Hc-2) where the superconductivity is finally destroyed. In this region is the vortex state, as the magnetic field enters the superconductor in a regular array of flux vortices (it is also known as the Abrikosov lattice from the Russian who first predicted the state).

De Haas-van Alphen oscillations have been used for many years to study the Fermi surface of electrons. They are the result of the quantization of energy levels of electrons which orbit in a magnetic field. The level spacing is proportional to the magnetic field, and the number of electrons that can be in any one level increases as the orbits get smaller, which also corresponds to an increasing magnetic field. So at small magnetic fields, the electrons are in a large number of closely spaced levels, while in large fields they are in a small number of widely spaced levels. Between these extremes, as the field is increased, the levels all move up in energy, while the uppermost filled level loses its electrons. The energy of the system oscillates during the process, the oscillations being with regular period in the inverse magnetic field, and proportional to the area of the Fermi surface, which gives the details of the orbits of the highest energy electrons.

Observations have been made of de Haas - van Alphen Oscillations in the Vortex State, but with damped amplitude. A superconducting gap at the Fermi surface has been shown to reduce the amplitude of oscillations, and the existence of the vortex lattice is known to disturb the orbits of electrons (Landau orbits) which also reduces the amplitude of oscillations. Much theoretical work is in progress in an attempt to derive information about the details of the superconducting gap from the observed damping.

Strong Electron-Phonon Coupling Theory.

The original BCS theory of superconductivity, while giving a correct qualitative understanding of the mechanisms leading to superconductivity, contains a number of gross approximations which lead to incorrect estimates of quantities such as critical field and critical temperature, for a given value of the energy gap. The strong coupling theory of Migdal and Eliashberg goes beyond this, by including the phonon interactions in a more detailed manner, such that the electronic Green's functions (which indicate how the electrons move around) are renormalized (meaning that quantities such as the effective mass are changed - electrons "feel heavier" so are slowed down by interactions).

Fundamental Quantum Theory

The EPR paradox appears to show that certain "common sense" questions about an electron (or other "quantum" particle) have no meaning until an observation is made. Such a positivist interpretation is unsatisfactory, but generally ignored. The Dirac equation, which is the fundamental description of electron, as it is relativistic and produces the spin properties of electrons, contains an infinity of solutions which are of negative energy, and assumed to be occupied by an infinite number (or "Dirac sea") of electrons, which do not interact, but whose absence is observed as a "hole" or positron. This state of affairs may be considered unsatisfactory by Occham's rasor, if a simpler solution is at hand. Both problems may be resolved if a more time-symmetric theory is established, and boundary conditions in time are considered in an equivalent manner to boundary conditions in space, as Einstein's special relativity might suggest. See John Gribbin's interesting discussions.