BCM theory
Encyclopedia
BCM theory, BCM synaptic modification, or the BCM rule, named for Elie Bienenstock, Leon Cooper
, and Paul Munro, is a physical theory of learning in the visual cortex
developed in 1981. Due to its successful experimental predictions, the theory is arguably the most accurate model of synaptic plasticity
to date.
"The BCM model proposes a sliding threshold for Long-term potentiation
or Long-term depression
induction and states that synaptic plasticity is stabilized by a dynamic adaptation of the time-averaged postsynaptic activity. According to the BCM model, reducing the postsynaptic activity decreases the LTP threshold and increases the LTD threshold. The opposite applies to the increase in postsynaptic activity."
, theoretically capable of Turing complete computational complexity, and thus became a standard materialist model for the mind.
However, Hebb's rule has problems, namely that it has no mechanism for connections to get weaker and no upper bound for how strong they can get. In other words, the model is unstable, both theoretically and computationally. Later modifications gradually improved Hebb's rule, normalizing it and allowing for decay of synapses, where no activity or unsynchronized activity between neurons results in a loss of connection strength. New biological evidence brought this activity to a peak in the 1970s, where theorists formalized various approximations in the theory, such as the use of firing frequency instead of potential in determining neuron excitation, and the assumption of ideal and, more importantly, linear synaptic integration of signals. That is, there is no unexpected behavior in the adding of input currents to determine whether or not a cell will fire.
These approximations resulted in the basic form of BCM below in 1979, but the final step came in the form of mathematical analysis to prove stability and computational analysis to prove applicability, culminating in Bienenstock, Cooper, and Munro's 1982 paper.
Since then, experiments have shown evidence for BCM behavior in both the visual cortex
and the hippocampus
, the latter of which plays an important role in the formation and storage of memories. Both of these areas are well-studied experimentally, but both theory and experiment have yet to establish conclusive synaptic behavior in other areas of the brain. Furthermore, a biological mechanism for synaptic plasticity
in BCM has yet to be established.
where
is the synaptic weight of the 
th synapse, 
is that synapse's input current, 
is the weighted presynaptic output vector, 
is the postsynaptic activation function
that changes sign at some output threshold
, and 
is the (often negligible) time constant of uniform decay of all synapses.
This model is merely a modified form of the Hebbian learning rule,
, and requires a suitable choice of activation function, or rather, the output threshold, to avoid the Hebbian problems of instability. This threshold was derived rigorously in BCM noting that with 
and the approximation of the average output 
, for one to have stable learning it is sufficient that
or equivalently, that the threshold
, where 
and 
are fixed positive constants.
When implemented, the theory is often taken such that
where angle brackets are a time average and
is the time constant of selectivity.
The model has drawbacks, as it requires both long-term potentiation
and long-term depression
, or increases and decreases in synaptic strength, something which has not been observed in all cortical systems. Further, it requires a variable activation threshold and depends strongly on stability of the selected fixed points
and 
. However, the model's strength is that it incorporates all these requirements from independently-derived rules of stability, such as normalizability and a decay function with time proportional to the square of the output.
and LTD
in the hippocampus
. The data showed qualitative agreement with the final form of the BCM activation function. This experiment was later replicated in the visual cortex
, which BCM was originally designed to model. This work provided further evidence of the necessity for a variable threshold function for stability in Hebbian-type learning (BCM or others).
Experimental evidence has been non-specific to BCM until Rittenhouse et al. confirmed BCM's prediction of synapse modification in the visual cortex when one eye is selectively closed. Specifically,
where
describes the variance in spontaneous activity or noise in the closed eye and 
is time since closure. Experiment agreed with the general shape of this prediction and provided an explanation for the dynamics of monocular eye closure (monocular deprivation
) versus binocular eye closure. The experimental results are far from conclusive, but so far have favored BCM over competing theories of plasticity.
Leon Cooper
Leon N Cooper is an American physicist and Nobel Prize laureate, who with John Bardeen and John Robert Schrieffer, developed the BCS theory of superconductivity...
, and Paul Munro, is a physical theory of learning in the visual cortex
Visual cortex
The visual cortex of the brain is the part of the cerebral cortex responsible for processing visual information. It is located in the occipital lobe, in the back of the brain....
developed in 1981. Due to its successful experimental predictions, the theory is arguably the most accurate model of synaptic plasticity
Synaptic plasticity
In neuroscience, synaptic plasticity is the ability of the connection, or synapse, between two neurons to change in strength in response to either use or disuse of transmission over synaptic pathways. Plastic change also results from the alteration of the number of receptors located on a synapse...
to date.
"The BCM model proposes a sliding threshold for Long-term potentiation
Long-term potentiation
In neuroscience, long-term potentiation is a long-lasting enhancement in signal transmission between two neurons that results from stimulating them synchronously. It is one of several phenomena underlying synaptic plasticity, the ability of chemical synapses to change their strength...
or Long-term depression
Long-term depression
Long-term depression , in neurophysiology, is an activity-dependent reduction in the efficacy of neuronal synapses lasting hours or longer. LTD occurs in many areas of the CNS with varying mechanisms depending upon brain region and developmental progress...
induction and states that synaptic plasticity is stabilized by a dynamic adaptation of the time-averaged postsynaptic activity. According to the BCM model, reducing the postsynaptic activity decreases the LTP threshold and increases the LTD threshold. The opposite applies to the increase in postsynaptic activity."
Development
In 1949, Donald Hebb proposed a working mechanism for memory and computational adaption in the brain now called Hebbian learning, or the maxim that cells that fire together, wire together. This law formed the basis of the brain as the modern neural networkNeural network
The term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...
, theoretically capable of Turing complete computational complexity, and thus became a standard materialist model for the mind.
However, Hebb's rule has problems, namely that it has no mechanism for connections to get weaker and no upper bound for how strong they can get. In other words, the model is unstable, both theoretically and computationally. Later modifications gradually improved Hebb's rule, normalizing it and allowing for decay of synapses, where no activity or unsynchronized activity between neurons results in a loss of connection strength. New biological evidence brought this activity to a peak in the 1970s, where theorists formalized various approximations in the theory, such as the use of firing frequency instead of potential in determining neuron excitation, and the assumption of ideal and, more importantly, linear synaptic integration of signals. That is, there is no unexpected behavior in the adding of input currents to determine whether or not a cell will fire.
These approximations resulted in the basic form of BCM below in 1979, but the final step came in the form of mathematical analysis to prove stability and computational analysis to prove applicability, culminating in Bienenstock, Cooper, and Munro's 1982 paper.
Since then, experiments have shown evidence for BCM behavior in both the visual cortex
Visual cortex
The visual cortex of the brain is the part of the cerebral cortex responsible for processing visual information. It is located in the occipital lobe, in the back of the brain....
and the hippocampus
Hippocampus
The hippocampus is a major component of the brains of humans and other vertebrates. It belongs to the limbic system and plays important roles in the consolidation of information from short-term memory to long-term memory and spatial navigation. Humans and other mammals have two hippocampi, one in...
, the latter of which plays an important role in the formation and storage of memories. Both of these areas are well-studied experimentally, but both theory and experiment have yet to establish conclusive synaptic behavior in other areas of the brain. Furthermore, a biological mechanism for synaptic plasticity
Synaptic plasticity
In neuroscience, synaptic plasticity is the ability of the connection, or synapse, between two neurons to change in strength in response to either use or disuse of transmission over synaptic pathways. Plastic change also results from the alteration of the number of receptors located on a synapse...
in BCM has yet to be established.
Theory
The basic BCM rule takes the formwhere
is the synaptic weight of the th synapse, is that synapse's input current, is the weighted presynaptic output vector, is the postsynaptic activation functionActivation function
In computational networks, the activation function of a node defines the output of that node given an input or set of inputs. A standard computer chip circuit can be seen as a digital network of activation functions that can be "ON" or "OFF" , depending on input. This is similar to the behavior of...
that changes sign at some output threshold
, and is the (often negligible) time constant of uniform decay of all synapses.This model is merely a modified form of the Hebbian learning rule,
, and requires a suitable choice of activation function, or rather, the output threshold, to avoid the Hebbian problems of instability. This threshold was derived rigorously in BCM noting that with and the approximation of the average output , for one to have stable learning it is sufficient thator equivalently, that the threshold
, where and are fixed positive constants.When implemented, the theory is often taken such that
where angle brackets are a time average and
is the time constant of selectivity.The model has drawbacks, as it requires both long-term potentiation
Long-term potentiation
In neuroscience, long-term potentiation is a long-lasting enhancement in signal transmission between two neurons that results from stimulating them synchronously. It is one of several phenomena underlying synaptic plasticity, the ability of chemical synapses to change their strength...
and long-term depression
Long-term depression
Long-term depression , in neurophysiology, is an activity-dependent reduction in the efficacy of neuronal synapses lasting hours or longer. LTD occurs in many areas of the CNS with varying mechanisms depending upon brain region and developmental progress...
, or increases and decreases in synaptic strength, something which has not been observed in all cortical systems. Further, it requires a variable activation threshold and depends strongly on stability of the selected fixed points
and . However, the model's strength is that it incorporates all these requirements from independently-derived rules of stability, such as normalizability and a decay function with time proportional to the square of the output.Experiment
The first major experimental confirmation of BCM came in 1992 in investigating LTPLong-term potentiation
In neuroscience, long-term potentiation is a long-lasting enhancement in signal transmission between two neurons that results from stimulating them synchronously. It is one of several phenomena underlying synaptic plasticity, the ability of chemical synapses to change their strength...
and LTD
Long-term depression
Long-term depression , in neurophysiology, is an activity-dependent reduction in the efficacy of neuronal synapses lasting hours or longer. LTD occurs in many areas of the CNS with varying mechanisms depending upon brain region and developmental progress...
in the hippocampus
Hippocampus
The hippocampus is a major component of the brains of humans and other vertebrates. It belongs to the limbic system and plays important roles in the consolidation of information from short-term memory to long-term memory and spatial navigation. Humans and other mammals have two hippocampi, one in...
. The data showed qualitative agreement with the final form of the BCM activation function. This experiment was later replicated in the visual cortex
Visual cortex
The visual cortex of the brain is the part of the cerebral cortex responsible for processing visual information. It is located in the occipital lobe, in the back of the brain....
, which BCM was originally designed to model. This work provided further evidence of the necessity for a variable threshold function for stability in Hebbian-type learning (BCM or others).
Experimental evidence has been non-specific to BCM until Rittenhouse et al. confirmed BCM's prediction of synapse modification in the visual cortex when one eye is selectively closed. Specifically,
where
describes the variance in spontaneous activity or noise in the closed eye and is time since closure. Experiment agreed with the general shape of this prediction and provided an explanation for the dynamics of monocular eye closure (monocular deprivationMonocular deprivation
Monocular deprivation is an experimental technique used by neuroscientists to study central nervous system plasticity. Generally, one of an animal's eyes is sutured shut during a period of high cortical plasticity...
) versus binocular eye closure. The experimental results are far from conclusive, but so far have favored BCM over competing theories of plasticity.