Fig 1.
The calcitron model and calcium-based plasticity rules.
(A) Sources of Ca2+ at the synapse. Local glutamate release from an activated presynaptic axon binds to an NMDA receptor in the postsynaptic dendritic spine, enabling local Ca2+ influx. Depolarization of the neuron opens voltage-gated calcium channels (VGCCs), enabling calcium influx from global signals. (Glutamate also binds to AMPA receptors, enabling Na+ influx, and depolarization also affects NMDAR conductance.) (B) Possible sources of Ca2+ influx in a neuron. Ca2+ can enter due to presynaptic input (), heterosynaptically-induced depolarization of VGCCs (
), the backpropagating action potential (
) or a supervisory signal, such as a calcium plateau induced by input to the apical tuft (
) (Neuron image courtesy of Dean Geckt and MICrONs Consortium [80]). (C) The four Ca2+ sources in a point neuron model. Each Ca2+ source is associated with a respective coefficient (
determining how much Ca2+ comes from each source. (D) The calcium control hypothesis. [Ca2+] below
induces no change, [Ca2+] between
and
induces depression, and [Ca2+] above
induces potentiation. (E) Weight change as a function of calcium in the linear version of Ca2+-based plasticity, as in (D), shown as phase plane. Magnitude of weight change is independent of current weight. Blue indicates depression, red indicates potentiation, white indicates no change. (F) Step stimulus to show the plastic effect of different levels of
.
is either raised to a depressive level (
, blue line) or to a potentiative level (
, red line) for several timesteps, then reduced to 0. S and E refer to the start and end of the calcium step. (G) Dynamics of the linear rule in response to the step stimulus from (F). Synaptic weights increase or decrease linearly in response to the potentiative or depressive levels of calcium (red and blue traces, respectively), then remain stable after calcium is turned off. (H) Fixed points (black) and learning rates (pink) in the asymptotic fixed point – learning rate (FPLR) version of the calcium control hypothesis. (I) Weight change as a function of [Ca2+] for different values of the present synaptic weight. Darker colors indicate higher weights. (J) Phase plane of weight changes for the FPLR rule. (K) Stimulus to demonstrate FPLR rule, identical to F. (L) Dynamics of the FPLR rule. Synaptic weights potentiate or depress asymptotically toward the potentiative (wmax, 1) or depressive (wmin, 0) fixed point.
Fig 2.
Four kinds of Hebbian and anti-Hebbian learning using Ca2+.
(A1–A4) Different versions of Hebbian and anti-Hebbian learning rules are implemented by setting the respective coefficients (α and γ in Eq. (6)) for the local () and backpropagating spike-dependent (
) [Ca2+]. For each rule, we show the direction of plastic change (indicated by the letters above the bars: “N”: no change, “D”: depression, “P”: potentiation) for three different conditions: a synapse with active presynaptic input in the absence of a postsynaptic spike (“pre”), a synapse without local input in the presence of a postsynaptic spike (“post”) and at a synapse with active presynaptic input and a postsynaptic spike (“both”). The total synaptic [Ca2+] (
) for each condition is the sum of the local input-dependent [Ca2+] (
, green) and the spike-dependent [Ca2+] (
, pink). (When there is neither local input or a postsynaptic spike, the expected [Ca2+] is 0) (B1–B4) For each learning rule from (A1–A4), 10 random binary inputs (black: active, white: inactive) are presented to each synapse at each time step. (Inputs are identical for all learning rules). (C1–C4) Sum of weighted inputs at each time step for each learning rule shown in A1–A4 respectively. Dotted horizontal line indicates the spike threshold (
from Eq. (6)). Outputs that are above the threshold (produce a postsynaptic spike) are indicated by a red circle. (D1–D4) [Ca2+] per synapse for each time step for the 4 learning rules shown in A1–A4 respectively. (E1–E4) Bar codes indicating occurrence of potentiation (“P”, red), depression (“D”, blue) or no change (“N”, white) shown in A1–A4 respectively. (F1–F4) Synaptic weights over the course of the simulation for A1–A4 cases respectively.
Fig 3.
Possible plasticity rules for presynaptic input- and spike-dependent- calcium.
(A) First scenario for calcium thresholds, where the depressive region is larger than the pre-depressive region, i.e., . Setting different coefficient values (α and γ in Eq. (6)) for the local (
) and backpropagating spike-dependent (
) [Ca2+] can lead to thirteen possible learning rules in the case of binary input and output. Vertical lines indicate the values of α that would be needed to induce depression (blue line) or potentiation (red line) with presynaptic input alone, horizontal lines indicate the values of γ that would be required to induce plasticity with a postsynaptic spike alone, and diagonal lines indicate the values of α and γ that would induce plasticity at activated synapses in the presence of a postsynaptic spike. Asterisk indicates rule (DDD) that can’t be implemented under the alternative threshold scenario from panel (C). (B1–B13) Each of the 13 regions from panel (A) represented as a bar plot. (C) Second calcium-threshold scenario, where the pre-depressive region is larger than the depressive region, i.e.,
. Asterisk indicates rule (NNP) that can’t be implemented under the first threshold scenario from (A). (D) Bar plot for the NNP rule from panel (C). See S1 Fig for reversed plasticity thresholds, i.e., when
.
Fig 4.
Frequency-dependent pre- and post- synaptic plasticity in rate-based models.
(A) Calcium-dependent plasticity in a rate model where . In the absence of postsynaptic spikes, sufficiently strong presynaptic inputs (
) alone can generate plasticity (green bars), postsynaptic firing alone (
) can induce plasticity even at inactive synapses (pink bars), and the combination of presynaptic input and postsynaptic spiking can sum to induce plasticity. (B1–B5) [Ca2+] (binned into regions of no change (white), depressive (blue) or potentiative (red)) as a function of presynaptic (x-axis) and postsynaptic (y-axis) firing rate. Each panel has a different value for α ([Ca2+] per presynaptic spike) and
([Ca2+] per postsynaptic spike). Values for
and
as in A.
Fig 5.
Learning to recognize repetitive patterns with heterosynaptic plasticity.
(A) A “signal” pattern is presented repeatedly to the neuron interspersed with non-repeating random “noise” patterns of the same sparsity. (B) Within each input pattern (both signal or noise) inactive synapses depress (above at left) due to the heterosynaptic calcium, whereas active synapses will potentiate from the sum of heterosynaptic calcium,
, and local calcium
(above
at right) (C1) Signal and noise patterns are presented to the neuron. (S: signal, N: noise). (C2) Spiking output of the calcitron. Black: no spike, Red: spike. An ‘x’ marker indicates incorrect output (e.g., no spike in response to a signal pattern, or a spike in response to a noise pattern), filled circles indicate correct outputs. Note the increase in correct spiking output over time. (C3–C5) Calcium, plasticity, and weights over time respectively as the input patterns in C1 are presented.
Fig 6.
“One-shot flip-flop” (1SFF) plasticity.
(A1) Fixed points (, black line, left y-axis) and learning rates (
, pink line, right y-axis) for the different regions of
. For 1SFF learning, the learning rate is set to 1 in the depressive and potentiative regions of
for immediate switch-like plasticity. (A2) Exemplar stimulus illustrating plasticity dynamics. An instantaneous
pulse is generated at three timesteps over the course of the experiment. (A3) Synaptic weights over time in response to stimulus presented in A2. (B) 1SFF plasticity rule. Local input alone does not reach the depression threshold, a plateau potential alone induces a depressive
, but local input combined with a plateau potential induces potentiation. (C1) A repeated sequence of input patterns (0,1,2,3) corresponding to locations on a circular track that a mouse traverses at each timestep as it runs multiple laps. (C2) Externally generated supervisory signal (plateau potential), Z, presented at different locations over the course of the experiment. (C3–C5)
, plasticity, and weights, respectively for each time step. Synaptic inputs at the time of the supervisory signal are “written” to the synaptic weights at the following respective time step. (C6) Calcitron output at each time step. Red circles indicate spikes. Note that the neuron spikes at the location at which the supervisory signal occurred in the previous lap.
Fig 7.
Different mechanisms for homeostatic plasticity.
(A) Supervision circuit for homeostatic plasticity using only a potentiation supervisor. If the calcitron (“C”) fires above the target minimum rate (i.e., it activates an inhibitory population (“I”), which prevents the potentiation supervisor (
) from producing a supervisory signal. If the calcitron’s output falls below its target range (i.e.,
) the supervisor is disinhibited, sending a potentiative calcium signal (
) to the calcitron. (B) Plasticity rule for global homeostatic plasticity using an internal mechanism for depression and a circuit mechanism for potentiation, as in (A). Here
and
(B1) Overly strong outputs (
) produce sufficient calcium to depress all synapses; Overly weak outputs (
) result in the activation of
setting
, potentiating all synapses. Note that because this is a “global” strategy, the presynaptic input
does not affect the plasticity outcomes. (B2) Two input patterns (only even synapses or only odd synapses) are presented to the neuron in random order. (B3)
occurs whenever
. (B4–B6) [Ca2+], plasticity, and weights over the course of the simulation. Even-numbered synapses are initialized to low weights; odd-numbered synapses are initialized to large weights. (B7) Neural rate output. Blue ‘x’ indicates output below
(lower dashed line), red ‘x’ indicates output above
(upper dashed line), green circles indicate output in the acceptable range. (C) Plasticity rule for targeted homeostatic plasticity using the postsynaptic firing rate and the circuit from (A) as well as local [Ca2+]. Here, only synapses that are active when the firing rate is too low (even synapses) are potentiated, and only synapses that are activated when the firing rate is too high (odd synapses) are depressed, as the
is necessary to bring the
above the plasticity thresholds. (D) Supervision circuit for homeostatic plasticity using both a potentiation (
) and depression (
) supervisor. In addition to the disinhibitory circuit for the control of
as in (A), when the calcitron’s output is above the target output range (
), a depression supervisor (
) is activated, sending a depressive signal (
) to the calcitron. (E) Plasticity rule for global homeostatic plasticity using both
and
. Here, the postsynaptic spike-dependent calcium
is not used; only the supervisory calcium signals
are necessary. Note the different strengths of
and
in E1 and E3. (F) Plasticity rule for targeted homeostatic plasticity using
and
in combination with Clocalat active synapses, so even and odd synapses will be differentially potentiated and depressed.
Table 1.
Weight update in the standard perceptron learning rule.
Table 2.
Weight update in the asymptotic perceptron learning rule.
Fig 8.
Perceptron learning with the calcitron.
(A) Supervision circuit for perceptron learning using a “target” supervisor. Whenever the target label is 1, the supervisor sends a potentiative supervisory signal to the calcitron. (B1) Plasticity rule for the perceptron with a target supervisor. Note that an additional calcium threshold (dashed green line) for a post-potentiative neutral zone (PPNZ) where no plastic change occurs has been added to the plasticity rule. (B2) Six patterns, half of which are arbitrarily assigned to the positive class and half to the negative class (
or
, respectively, see tick labels on x-axis) are repeatedly presented to the calcitron in random order over several epochs. (B3) Supervisory signal. Appears whenever the target label
. (B4–B6) [Ca2+], plasticity, and weights over the course of the simulation. (B7) Calcitron output. Red circle: true positive, red ‘x’: false positive, black circle: true negative, black ‘x’: false negative. (C) Supervision circuit for perceptron learning using a “critic” supervisor. The supervisor compares the target label y to the calcitron output
. If the trial was a false negative (
), the supervisor sends a potentiative supervisory signal
to the calcitron. If the trial was a false positive (
), the supervisor sends a depressive supervisory signal
to the calcitron. (D1–D7) Perceptron learning with the “critic” supervisory circuit. Note the different magnitudes of the supervisory signal in (D3) – the large signal corresponds to
and the small signal corresponds to
.
Table 3.
Simulation parameters.