Quasi-stable attractor states as a framework for neural computing

Above: Activity of a simple model network with randomly connected groups of neurons. Mean activity of each group is color-coded according to the scale on the right. Noise causes frequent transitions between quasistable states, each state containing a distinct set of active units. Similar state sequences are extracted by hidden Markov modeling from in vivo data accrued from animals behaving in a variety of manners.

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.

Images producing bistable percepts. A) The Necker cube can be seen with its front face as bottom-left or top-right. The 3D percept produced by such a 2D line-drawing is experience-dependent: a 50-year old who gained sight after being blind from birth did not perceive a cube. B) Rubin's vase can appear as a single vase in the center or two black faces looking at each other (first created by the Danish psychologist Edgar Rubin in 1915). C) The image by Joseph Jastrow in 1899 can appear either as a leftward facing duck or a rightward facing rabbit.

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

  • Click here for work on a decision-making model.
  • 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.

  • Click here for computer code of the discrimination model.
  • 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.

    Last modified November 2, 2006

    Paul Miller, pmiller@brandeis.edu

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