Abstract

Perception of supraliminal stimuli might in general be reflected in bursts of action potentials (spikes), and their memory traces could be formed through spike-timing-dependent plasticity (STDP). Memory traces for subliminal stimuli might be formed in a different manner, because subliminal stimulation evokes a fraction (but not a burst) of spikes. Simulations of a cortical neural network model showed that a subliminal stimulus that was too brief (10 msec) to perceive transiently (more than about 500 msec) depolarized stimulus-relevant principal cells and hyperpolarized stimulus-irrelevant principal cells in a subthreshold manner. This led to a small increase or decrease in ongoing-spontaneous spiking activity frequency (less than 1 Hz). Synaptic modification based on STDP during this period effectively enhanced relevant synaptic weights, by which subliminal learning was improved. GABA transporters on GABAergic interneurons modulated local levels of ambient GABA. Ambient GABA molecules acted on extrasynaptic receptors, provided principal cells with tonic inhibitory currents, and contributed to achieving the subthreshold neuronal state. We suggest that ongoing-spontaneous synaptic alteration through STDP following subliminal stimulation may be a possible neuronal mechanism for leaving its memory trace in cortical circuitry. Regulation of local ambient GABA levels by transporter-mediated GABA import and export may be crucial for subliminal learning.

1  Introduction

The brain is highly adaptive and can adjust rapidly to surrounding environments. Perceptual learning is one of such adaptive processes. For instance, our sensitivity to a stimulus we previously encountered could be improved (Zohary, Celebrini, Britten, Newsome, 1994; Gilbert, 1996; Vaina, Belliveau, des Roziers, Zeffiro, 1998; Grossberg, 1999). It is often claimed that such an improvement in perceptual ability arises from alteration of cortical circuitry (Gilbert, 1994). Karni and Sagi (1991) have proposed a Hebbian process for synaptic alteration in which use-dependent synaptic enhancement is induced by bursts of concurrent pre- and postsynaptic action potentials (spikes). Spike-timing-dependent plasticity (STDP) determines whether synapses are potentiated or depressed, by which cortical circuitry is reorganized in an experience-dependent manner (Bi & Poo, 1998; Young et al., 2007).

Learning without perception, called subliminal learning, is known to influence perceptual processes as well. For example, a brief exposure to an item will increase a subject’s preference for that item (for review, see Hirst, 1997). The level of stimulation is below a perceptual threshold—too brief or weak for subjects to perceive. This implies that subliminal learning is an unconscious process. Interestingly, the information of subliminally presented sensory stimuli can be maintained not just for a few seconds but also for minutes or more; that is, they can become long-term memories.

Kern and Shaker (2002) recorded functional magnetic resonance imaging (fMRI) to subliminal rectal distention in human subjects. Although lower in intensity and volume, cortical activation caused by the subliminal stimulation was similar to that by supraliminal stimulation. Brázdil and colleagues (Brázdil, Rektor, Jurák, & Daniel, 1998) investigated the relationship between subliminal stimulation and electrical activity in the visual system. They demonstrated that subliminal stimulation had an impact on neuronal activity, showing a weak but significant increase in amplitude of cortical EEG signals. Since the EEG signal reflects collective neuronal behavior, its increase implies that the neuronal activation induced by the subliminal stimulation might synchronize. Shevrin and colleagues (Shevrin, 2001; Bernat, Bunce, & Shevrin, 2001) demonstrated that although smaller in amplitude, event-related potentials induced by subliminal visual stimulation had a component structure similar to that by supraliminal stimulation.

These studies raise several important questions. How does subliminal learning leave a memory trace for unconscious information processing? Can the brain reliably carry out synaptic modification based on STDP, when subliminal stimulation merely induces a subthreshold (but not a suprathreshold) neuronal response, as is reflected in such a fraction (but not a burst) of spikes? To answer these questions, we simulate a cortical neural network model. In our model, two types of GABAergic interneurons (Ia and Ib) are employed. Transporters on axon terminal membranes of Ia and Ib cells modulate ambient GABA concentrations around their (axonal) target cells. Ambient GABA molecules act on extrasynaptic receptors and provide these target cells with inhibitory currents in a tonic manner (Brickley & Mody, 2012). When exposed to a sensory stimulus, synaptic connection weights between principal cells are modified in a spike-timing-dependent manner. This hypothetical treatment was proposed for synaptic alteration in our previous study (Hoshino 2011b). Statistically analyzing neuronal activities (subthreshold membrane potentials and spikes), we try to elucidate how memory traces of subliminal stimuli could be formed in the cortex.

2  Neural Network Model

2.1  Structure

A neural network model of a primary sensory cortical area is shown in Figure 1. It responds to particular sensory features fn (). Activation of a population of neurons (i.e., a dynamic cell assembly) encodes a single feature. Cell assemblies comprise cell units (see the gray circle): one principal cell (P) and two GABAergic interneurons (Ia and Ib). Within cell assemblies, P cells are recurrently connected via excitatory synapses. An Ia cell receives an excitatory projection from a P cell and sends an inhibitory projection to an Ib cell. An Ib cell receives excitatory projections from P cells belonging to different cell assemblies (), and sends an inhibitory projection to a P cell belonging to the same assembly (). This circuitry gives the so-called lateral inhibitory mechanism between cell assemblies. P cells belonging to cell assembly n receive input current when the network is presented with sensory feature fn.

Figure 1:

Neural network model. Cell assemblies () consist of cell units (see the gray circle). P, Ia, and Ib denote a principal cell and GABAergic interneurons, respectively. When presented with a feature stimulus fn, P cells belonging to cell assembly n receive a sensory input current. Inset: A schematic illustration for hypothetical modulation of ambient GABA concentration. An example is shown for an Ib-to-P projection. The gray area points to an extracellular space for local ambient GABA concentration around a P cell that is modulated depending on presynaptic Ib cell activity (Hoshino, 2009), being increased or decreased from basal ambient GABA concentration.

Figure 1:

Neural network model. Cell assemblies () consist of cell units (see the gray circle). P, Ia, and Ib denote a principal cell and GABAergic interneurons, respectively. When presented with a feature stimulus fn, P cells belonging to cell assembly n receive a sensory input current. Inset: A schematic illustration for hypothetical modulation of ambient GABA concentration. An example is shown for an Ib-to-P projection. The gray area points to an extracellular space for local ambient GABA concentration around a P cell that is modulated depending on presynaptic Ib cell activity (Hoshino, 2009), being increased or decreased from basal ambient GABA concentration.

We employ a conductance-based, integrate-and-fire neuron model with a GABA transport mechanism that was proposed in our previous study (Hoshino, 2009). We briefly explain the mechanism. As suggested by Richerson and colleagues (Richerson & Wu, 2003; Wu, Wang, & Richerson, 2003; Richerson, 2004; Wu, Wang, Diez-Sampedro, Richerson, 2007), a transporter such as GAT-1, embedded in neuronal (and glial) membranes, is not simply an importer removing GABA from the extracellular space. Instead, it operates in an ion-coupled manner, pursuing an equilibrium point that is determined by the stoichiometry of transporter, the concentration gradients of substrates, and the membrane potential. Under normal physiological conditions, a thermodynamic reaction cycle involves coupled translocation of two ions, one ion, and one uncharged GABA molecule. The cotransported molecules (2, , GABA) cross the membrane together. The driving force for the coupled transport is the electrochemical potential, which is the sum of the electropotential and the chemical potential.

The reversal potential of transporter is the equilibrium membrane voltage at which the value of electrochemical potential is equal to 0. Under the normal physiological condition, the reversal potential is close to the resting potential of neurons. At membrane potentials below the reversal potential, net influx of GABA, called forward transport or GABA import, takes place. When membrane potentials are above the reversal potential, net efflux of GABA, called reverse transport or GABA export, takes place. In our model, GABA transporters, embedded in axon terminal membranes of Ia and Ib cells, regulate local ambient GABA levels around their (axonal) target cells. An example is shown in the inset of Figure 1 for an Ib-to-P projection. Transporters, which are embedded in the axon terminal membrane of an Ib cell, import (remove) GABA from the extracellular space when the Ib cell is hyperpolarized, leading to a decrease in ambient GABA concentration around a P cell (see the gray area). GABA is exported into the extracellular space if the Ib cell is depolarized, leading to an increase in ambient GABA concentration. Namely, presynaptic terminals work for the source of GABA. Ambient GABA molecules are accepted by extrasynaptic GABAa receptors and tonically inhibit the P cell. The model description is given in appendixes  A to  D and Table 1.

Table 1:
List of Parameters and Their Values.
DescriptionParameterValue
Membrane capacitance of type Y (Y = P, Ia, Ib) cell   
Membrane conductance   
Resting potential     
Maximal conductance for type Z (AMPA, GABA) receptor   
Reversal potential   
Number of cell units within cell assemblies   
Number of cell assemblies   
Synaptic weight (strength) from jth to ith P cell within cell assembly n at time t   
Synaptic weight from ith Ib to P cell   
Synaptic weight from ith P to Ia cell   
Synaptic weight from jth (cell assembly k) P to ith (cell assembly n) Ib cell    
Synaptic weight from ith Ia to P cell   in Figure 12B; otherwise 0 
Synaptic weight from ith Ia to Ib cell   
Amount of extrasynaptic GABAa receptors on type Y (Y = P, Ia, Ib) cell   
Intensity of input   
Channel opening rate for type Z (AMPA, GABA) receptor   
   
Channel closing rate   
Concentration of glutamate in the synaptic cleft released from P cell   
Concentration of GABA in the synaptic cleft released from type X (Ia, Ib) cell   
Decay constant for ambient GABA concentration   
Basal ambient GABA concentration   
GABA transfer coefficient for type X (Ia, Ib) cell   
Maximal ambient GABA concentration   
Minimal ambient GABA concentration   
Reversal potential of transporter for type X (Ia, Ib) cell   
Steepness of sigmoid function for type Y (Y = P, Ia, Ib) cell   
Threshold of sigmoid function   
Synaptic decay rate   
Synaptic modification rate   
Built-in synaptic weight     
Potentiation rate   
Depression rate   
Potentiation time constant   
Depression time constant   
DescriptionParameterValue
Membrane capacitance of type Y (Y = P, Ia, Ib) cell   
Membrane conductance   
Resting potential     
Maximal conductance for type Z (AMPA, GABA) receptor   
Reversal potential   
Number of cell units within cell assemblies   
Number of cell assemblies   
Synaptic weight (strength) from jth to ith P cell within cell assembly n at time t   
Synaptic weight from ith Ib to P cell   
Synaptic weight from ith P to Ia cell   
Synaptic weight from jth (cell assembly k) P to ith (cell assembly n) Ib cell    
Synaptic weight from ith Ia to P cell   in Figure 12B; otherwise 0 
Synaptic weight from ith Ia to Ib cell   
Amount of extrasynaptic GABAa receptors on type Y (Y = P, Ia, Ib) cell   
Intensity of input   
Channel opening rate for type Z (AMPA, GABA) receptor   
   
Channel closing rate   
Concentration of glutamate in the synaptic cleft released from P cell   
Concentration of GABA in the synaptic cleft released from type X (Ia, Ib) cell   
Decay constant for ambient GABA concentration   
Basal ambient GABA concentration   
GABA transfer coefficient for type X (Ia, Ib) cell   
Maximal ambient GABA concentration   
Minimal ambient GABA concentration   
Reversal potential of transporter for type X (Ia, Ib) cell   
Steepness of sigmoid function for type Y (Y = P, Ia, Ib) cell   
Threshold of sigmoid function   
Synaptic decay rate   
Synaptic modification rate   
Built-in synaptic weight     
Potentiation rate   
Depression rate   
Potentiation time constant   
Depression time constant   

2.2  Parameter Setting

The Ia and Ib cells may correspond to fast- and late-spiking cells such as small basket cells and chandelier cells for the former and large basket cells for the latter (Kawaguchi, 1995; Kawaguchi & Kondo, 2002). The parameters for each cell type are given in Table 1. The fast-spiking cell has a small membrane time constant (10 msec), and the late-spiking cell a large membrane time constant (30 msec). In this model, the fast Ia cell induces rapid inhibition of Ib cells, generating modulation of ambient GABA levels as soon as a subliminal stimulus is presented.

To make a weak but significant response to the subliminal stimulus, it is necessary for principal cells to oscillate near threshold for their spike generation. Such a subthreshold neuronal state is reflected in sparse, ongoing-spontaneous firing (i.e., background firing), which is achieved by carefully adjusting network and neuron parameters. Mutual excitation among principal cells (via their recurrent excitatory synaptic connections; see in Table 1) is the driver for the generation of background firing. Control of a global level of ambient GABA (see in Table 1) is also important. Elevating ambient GABA concentration leads to an increase in tonic inhibitory current in principal cells via extrasynaptic GABA receptors and thus to the suppression of network activity, while its lowering leads to a decrease in inhibitory current and thus to the excitation of network activity.

Figure 2A shows the dependence of ongoing-spontaneous spiking activity (top) and membrane potential (bottom) recorded from a principal (P) cell on the recurrent (P-to-P) connection weight: (see equation A.2 in appendix  A). No external excitatory current was applied. The P cell occasionally fires even without the recurrent connection (; see Table 1) because of a stochastic spike generation scheme (see equation C.1 in appendix  C). The background spiking activity is increased as the weight increases (see the traces in the top panel). Namely, the background firing is driven through recurrent (P-to-P) excitation. The increase of ongoing-spontaneous spiking activity arises from a slight membrane depolarization (see the traces in the bottom panel).

Figure 2:

Ongoing-spontaneous, subthreshold membrane potential oscillation. (A) Dependence of ongoing-spontaneous spiking activity (top) and membrane potential (bottom) recorded from a principal (P) cell on the recurrent (P-to-P) connection weight ; see equation A.2 in appendix  A. No external excitatory current was applied, and no learning process took place. (B) Relationships of ongoing-spontaneous spiking activity (top: average firing rate) and membrane potential (bottom: average membrane potential) to the recurrent connection weight. The arrow points to an optimal network state at which P cells reliably oscillate near firing threshold.

Figure 2:

Ongoing-spontaneous, subthreshold membrane potential oscillation. (A) Dependence of ongoing-spontaneous spiking activity (top) and membrane potential (bottom) recorded from a principal (P) cell on the recurrent (P-to-P) connection weight ; see equation A.2 in appendix  A. No external excitatory current was applied, and no learning process took place. (B) Relationships of ongoing-spontaneous spiking activity (top: average firing rate) and membrane potential (bottom: average membrane potential) to the recurrent connection weight. The arrow points to an optimal network state at which P cells reliably oscillate near firing threshold.

Figure 2B shows the relationships of ongoing-spontaneous spiking activity (top: average firing rate) and membrane potential (bottom: average membrane potential) to the recurrent connection weight. The arrow points to an optimal network state (; see Table 1) at which P cells reliably oscillate near firing threshold. For , burst firing often occurred even without sensory stimulation (not shown).

Figure 3 shows the dependence of ongoing-spontaneous spiking activity (top) and membrane potential (bottom) recorded from a P cell on the basal ambient GABA concentration (see in equation B.4 in appendix  B). The background spiking activity is decreased as the basal ambient GABA concentration increases (see the traces in the top panel), indicating that the global level of ambient GABA is important for tuning the background firing state. The decrease of ongoing-spontaneous spiking activity arises from a slight membrane hyperpolarization (see the traces in the bottom panel). (see Table 1) yields a reliable membrane potential oscillation near firing threshold in P cells.

Figure 3:

Ongoing-spontaneous, subthreshold membrane potential oscillation regulated by ambient GABA. Dependence of ongoing-spontaneous spiking activity (top) and membrane potential (bottom) recorded from a principal (P) cell on the basal ambient GABA concentration ; see equation B.4 in appendix  B. No external excitatory current was applied and no learning process took place.

Figure 3:

Ongoing-spontaneous, subthreshold membrane potential oscillation regulated by ambient GABA. Dependence of ongoing-spontaneous spiking activity (top) and membrane potential (bottom) recorded from a principal (P) cell on the basal ambient GABA concentration ; see equation B.4 in appendix  B. No external excitatory current was applied and no learning process took place.

3  Results

3.1  Subliminal Learning Through Ongoing-Spontaneous Spike-Timing-Dependent Plasticity

As shown in Figure 4A, bursts of spikes are evoked in P cells (see n = 2) when the network is presented with a supraliminal stimulus (f2; see the left horizontal line). Then we applied a subliminal stimulus (see the center, short horizontal line). It was so brief (10 msec) that a fraction (0–2 spikes in each cell) but not a burst of spikes was evoked. A learning process was carried out based on STDP, by which synaptic weights between P cells were modified (see appendix  D). Note that the learning period involved the ongoing-spontaneous activity time period following the subliminal stimulation. Henceforth, we call this ongoing-spontaneous STDP (OS-STDP). After the learning, we applied the same supraliminal stimulus (see the right horizontal line). We found that the learning can enhance the responsiveness of the network to that supraliminal stimulus.

Figure 4:

Subliminal learning. (A) Ongoing-spontaneous spike-timing-depen-dent plasticity (OS-STDP). Left: Raster plots of action potentials evoked in P cells belonging to respective cell assemblies (). A supraliminal stimulus (f2: input) was presented to the network (see the left horizontal line). When presented with a subliminal stimulus (see the center, short horizontal line), a learning process (learning) took place and synaptic weights between P cells were modified. The same supraliminal stimulus was presented after the learning (see the right horizontal line). Right: Histogram of synaptic weight between stimulus-relevant P cells after the learning. The initial weight was set to 4 (see in equation A.2 in appendix  A and Table 1). (B) One-shot spike-timing-dependent plasticity (one-shot STDP). Raster plots of action potentials (left) and synaptic weights (right). The learning process was restricted to the subliminal stimulation period (learning). (C) One-shot STDP enhanced. The synaptic modification rate (see in equation D.1 in appendix  D and Table 1) was increased from 3 to 8. (D) Cumulative spike counts for the second supraliminal stimulation. The solid and dashed traces denote OS-STDP (see panel A) and one-shot STDP enhanced (see panel C), respectively.

Figure 4:

Subliminal learning. (A) Ongoing-spontaneous spike-timing-depen-dent plasticity (OS-STDP). Left: Raster plots of action potentials evoked in P cells belonging to respective cell assemblies (). A supraliminal stimulus (f2: input) was presented to the network (see the left horizontal line). When presented with a subliminal stimulus (see the center, short horizontal line), a learning process (learning) took place and synaptic weights between P cells were modified. The same supraliminal stimulus was presented after the learning (see the right horizontal line). Right: Histogram of synaptic weight between stimulus-relevant P cells after the learning. The initial weight was set to 4 (see in equation A.2 in appendix  A and Table 1). (B) One-shot spike-timing-dependent plasticity (one-shot STDP). Raster plots of action potentials (left) and synaptic weights (right). The learning process was restricted to the subliminal stimulation period (learning). (C) One-shot STDP enhanced. The synaptic modification rate (see in equation D.1 in appendix  D and Table 1) was increased from 3 to 8. (D) Cumulative spike counts for the second supraliminal stimulation. The solid and dashed traces denote OS-STDP (see panel A) and one-shot STDP enhanced (see panel C), respectively.

As shown in Figure 4B, the responsiveness cannot not be enhanced if the learning is restricted to the subliminal stimulation period, that is, if the following ongoing-spontaneous activity time period is excluded from the learning period. Henceforth, we call this one-shot STDP. These results indicate that synaptic alteration during the ongoing-spontaneous activity time period following subliminal stimulation is crucial for leaving its memory trace in cortical circuitry. The right panels present histograms of synaptic weights between stimulus-relevant P cells after the learning through OS-STDP (see Figure 4A) or one-shot STDP (see Figure 4B).

As shown in Figure 4C, one-shot STDP (see Figure 4B) can be enhanced if the synaptic modification rate (see in equation D.1 of appendix  D) is extremely increased. However, as shown in Figure 4D, one-shot STDP (see the dashed trace) cannot overwhelm OS-STDP (see the solid trace) in the enhancement of the responsiveness. Note that the synaptic weights (averaged) after the learning for both conditions (OS-STDP and one-shot STDP enhanced) are almost identical, but their distributions are considerably different (compare Figures 4A and 4C, right).

Figure 5A shows burst activity when presented with a supraliminal stimulus (see the top trace), missing burst activity when presented with a subliminal stimulus (see the middle trace), and ongoing-spontaneous (i.e., without stimulation) activity (see the bottom trace). Note that no learning process took place. We counted the number of spikes during a given time period (1 sec; see the long horizontal line for the supraliminal stimulus) at different stimulus-onset times (N = 10). Figure 5B (top) statistically shows the burst activity (“supra.”), the missing burst activity (“sublim.”), and the ongoing-spontaneous activity (“ong.-spont.”). Figure 5B (bottom) shows spikes evoked by the supraliminal (supra.) and subliminal (sublim.) stimuli, which are indicated as “supra.” minus ong.-spont. and sublim. minus ong.-spont., respectively (see the top panel). The missing burst activity is slightly higher than the ongoing-spontaneous activity (see the open circle, “sublim”): less than 1 spike/sec on average. The subliminal stimulation causes a small decrease in stimulus-irrelevant P cell activity (see the filled circle, sublim.).

Figure 5:

Spikes evoked by supraliminal and subliminal stimulation. (A) Top: Burst activity when presented with a supraliminal stimulus. Middle: Missing burst activity when presented with a subliminal stimulus. Bottom: ongoing-spontaneous (i.e., without stimulation) activity. No learning process took place. (B) Statistical evaluation of spike generation. Top: The burst activity (supra.), the missing burst activity (sublim.), and the ongoing-spontaneous activity (ong.-spont.) with meanSD. The number of spikes was counted during a given time period (1 sec; see the long horizontal line for the supraliminal stimulus in panel A) at different stimulus-onset times (N = 10). Bottom: Spikes evoked by the supraliminal (supra.) and subliminal (sublim.) stimuli, which are indicated as supra. minus ong.-spont. and sublim. minus ong.-spont., respectively (see the top panel). The open and filled circles denote stimulus-relevant (n = 2) and stimulus-irrelevant (n = 1) P cells.

Figure 5:

Spikes evoked by supraliminal and subliminal stimulation. (A) Top: Burst activity when presented with a supraliminal stimulus. Middle: Missing burst activity when presented with a subliminal stimulus. Bottom: ongoing-spontaneous (i.e., without stimulation) activity. No learning process took place. (B) Statistical evaluation of spike generation. Top: The burst activity (supra.), the missing burst activity (sublim.), and the ongoing-spontaneous activity (ong.-spont.) with meanSD. The number of spikes was counted during a given time period (1 sec; see the long horizontal line for the supraliminal stimulus in panel A) at different stimulus-onset times (N = 10). Bottom: Spikes evoked by the supraliminal (supra.) and subliminal (sublim.) stimuli, which are indicated as supra. minus ong.-spont. and sublim. minus ong.-spont., respectively (see the top panel). The open and filled circles denote stimulus-relevant (n = 2) and stimulus-irrelevant (n = 1) P cells.

Figures 6A and 6B present stimulus-evoked activity (top), average synaptic weight (middle), and average ongoing-spontaneous membrane potential (bottom) obtained for OS-STDP and one-shot STDP, respectively, as a function of the synaptic modification rate (; see equation D.1 in appendix  D and Table 1). Figure 6C presents relationships of the stimulus-evoked activity (left) and the average ongoing-spontaneous membrane potential (right) to the average synaptic weight. They were derived from Figures 6A and 6B. These results indicate that OS-STDP can enhance the responsiveness of P cells with smaller synaptic enhancement (see the open squares) compared to one-shot STDP (see the filled squares). Slight ongoing-spontaneous membrane depolarization (see the open inverted triangles) might be essential for the sensory enhancement because it corresponds functionally to a decrease in firing threshold.

Figure 6:

Neuronal and synaptic modulation through STDP. (A) OS-STDP took place. (B) One-shot STDP took place. Stimulus-evoked activity (top), average synaptic weight (middle), and average ongoing-spontaneous membrane potential (bottom) are shown as a function of the synaptic modification rate: . (C) Relationships of the stimulus-evoked activity (left) and the average ongoing-spontaneous membrane potential (right) to the average synaptic weight. They were derived from those in panels A and B. The open and filled symbols denote OS-STDP and one-shot STDP, respectively.

Figure 6:

Neuronal and synaptic modulation through STDP. (A) OS-STDP took place. (B) One-shot STDP took place. Stimulus-evoked activity (top), average synaptic weight (middle), and average ongoing-spontaneous membrane potential (bottom) are shown as a function of the synaptic modification rate: . (C) Relationships of the stimulus-evoked activity (left) and the average ongoing-spontaneous membrane potential (right) to the average synaptic weight. They were derived from those in panels A and B. The open and filled symbols denote OS-STDP and one-shot STDP, respectively.

Figure 7A shows in more detail how the ongoing-spontaneous membrane depolarization occurs in stimulus-relevant P cells (see the solid trace, n = 2). Note that the depolarization does not reach firing threshold. Namely, OS-STDP modulates the P cells in a subthreshold manner. As shown in Figure 7B, such an ongoing-spontaneous subthreshold depolarization is caused by a small decrease in ambient GABA concentration around the stimulus-relevant P cells (see solid trace, n = 2).

Figure 7:

Modulation of ongoing-spontaneous neuronal activity by subliminal learning. (A) Ongoing-spontaneous membrane potentials recorded from stimulus-relevant (see traces n = 2) and stimulus-irrelevant (see traces n = 1 and n = 3) P cells long (10 seconds) after the subliminal learning. OS-STDP (see the solid traces) or one-shot STDP (see the dashed traces) took place. We got the same solution for the differential equation (see equation A.1 in appendix  A) because of the same initial values employed. OS-STDP and one-shop STDP slightly change the solution, that is, the ongoing-spontaneous membrane potential below firing threshold. This leads to highly synchronizing subthreshold membrane potential oscillations; compare the solid and dashed traces. (B) Modulation of ongoing-spontaneous ambient GABA concentrations by subliminal learning. Ambient GABA concentrations around stimulus-relevant (see traces n = 2) and stimulus-irrelevant (see traces n = 1 and n = 3) P cells are shown. OS-STDP (solid traces) or one-shot STDP (dashed traces) took place.

Figure 7:

Modulation of ongoing-spontaneous neuronal activity by subliminal learning. (A) Ongoing-spontaneous membrane potentials recorded from stimulus-relevant (see traces n = 2) and stimulus-irrelevant (see traces n = 1 and n = 3) P cells long (10 seconds) after the subliminal learning. OS-STDP (see the solid traces) or one-shot STDP (see the dashed traces) took place. We got the same solution for the differential equation (see equation A.1 in appendix  A) because of the same initial values employed. OS-STDP and one-shop STDP slightly change the solution, that is, the ongoing-spontaneous membrane potential below firing threshold. This leads to highly synchronizing subthreshold membrane potential oscillations; compare the solid and dashed traces. (B) Modulation of ongoing-spontaneous ambient GABA concentrations by subliminal learning. Ambient GABA concentrations around stimulus-relevant (see traces n = 2) and stimulus-irrelevant (see traces n = 1 and n = 3) P cells are shown. OS-STDP (solid traces) or one-shot STDP (dashed traces) took place.

3.2  Modulation of Ongoing-Spontaneous Neuronal Activity by Subliminal Stimulation

Figure 8A (see the solid traces) shows how the subliminal stimulation modulates ongoing-spontaneous membrane potentials of stimulus-relevant (n = 2) and stimulus-irrelevant (n = 1 and n = 3) P cells. The dashed traces represent ongoing-spontaneous membrane potentials without stimulation. Figure 8B presents differences between these membrane potentials (i.e., between the solid and dashed traces in Figure 8A), indicating that the subliminal stimulation slightly depolarizes stimulus-relevant P cells (see trace n = 2) and hyperpolarizes stimulus-irrelevant P cells (see trace n = 1 and n = 3) for more than about 500 msec following the subliminal stimulation. We recorded membrane potentials at arbitrary stimulus-onset times (N = 10) and calculated their differences as those in panels A and B. Figure 8C shows the integration of these differences. This result indicates statistically significant depolarization of stimulus-relevant principal (P) cells (n = 2) and hyperpolarization of stimulus-irrelevant P cells (n = 1 and n = 3) induced by the subliminal stimulation.

Figure 8:

Transient, slight modulation of ongoing-spontaneous membrane activity by subliminal stimulation. (A) Ongoing-spontaneous membrane potentials recorded from stimulus-relevant (see traces n = 2) and stimulus-irrelevant (see traces n = 1 and n = 3) P cells after subliminal stimulation (input: f2). The subliminal stimulus was presented (solid traces) or not (dashed traces). No learning process took place. (B) Differences in ongoing-spontaneous membrane potential between these two conditions (with and without stimulation) shown in panel A. The positive and negative values indicate that the subliminal stimulus (input: f2) induces a depolarization (see trace n = 2) and a hyperpolarization (see trace n = 1, 3), respectively, during the following ongoing-spontaneous activity time period. (C) Statistical evaluation of membrane depolarization and hyperpolarization. Membrane potentials were recorded at arbitrary stimulus-onset times (N = 10) and their differences were calculated as in panels A and B. These differences were integrated (see the circles).

Figure 8:

Transient, slight modulation of ongoing-spontaneous membrane activity by subliminal stimulation. (A) Ongoing-spontaneous membrane potentials recorded from stimulus-relevant (see traces n = 2) and stimulus-irrelevant (see traces n = 1 and n = 3) P cells after subliminal stimulation (input: f2). The subliminal stimulus was presented (solid traces) or not (dashed traces). No learning process took place. (B) Differences in ongoing-spontaneous membrane potential between these two conditions (with and without stimulation) shown in panel A. The positive and negative values indicate that the subliminal stimulus (input: f2) induces a depolarization (see trace n = 2) and a hyperpolarization (see trace n = 1, 3), respectively, during the following ongoing-spontaneous activity time period. (C) Statistical evaluation of membrane depolarization and hyperpolarization. Membrane potentials were recorded at arbitrary stimulus-onset times (N = 10) and their differences were calculated as in panels A and B. These differences were integrated (see the circles).

As shown in Figure 9A, the slight membrane depolarization induced by the subliminal stimulus (see trace n = 2 in Figure 8B) leads to a small increase in ongoing-spontaneous spiking activity in stimulus-relevant P cells (see the dashed squares within which open circles emerge). The emergence of open circles implies that action potentials are evoked by the subliminal stimulus. The small but significant increase in ongoing-spontaneous spiking activity allows the synaptic weights to be enhanced through OS-STDP (see Figure 4A, right). On the other hand, the slight membrane hyperpolarization induced by the subliminal stimulus (see trace n = 1 and n = 3 in Figure 8B) leads to a small decrease in ongoing-spontaneous spiking activity in stimulus-irrelevant P cells (see the dashed ovals within which open circles disappear). The disappearance of open circles implies that action potentials are eliminated by the subliminal stimulus. This avoids interference with subliminal learning. (For its statistical evaluation, see Figure 5B.)

Figure 9:

Transient, slight modulation of ongoing-spontaneous spiking activity and ambient GABA concentration by subliminal stimulation. (A) Spiking activity. The open circles denote action potentials in P cells when presented with a subliminal stimulus (f2). The filled circles denote those without stimulation. The ongoing-spontaneous spiking activity is transiently (more than about 500 msec) slightly modulated by the subliminal stimulus (see the dashed squares and ovals). No learning process took place. Inside the dashed squares, the emergence of open circles indicates a small increase in ongoing-spontaneous spiking activity, and their disappearance inside the dashed ovals indicates a small decrease in ongoing-spontaneous spiking activity; see also Figure 5B. (B) Ambient GABA concentrations around stimulus-relevant (see trace n = 2) and stimulus-irrelevant (see trace n = 1 and n = 3) P cells. The subliminal stimulus (f2) was presented (solid traces) or not (dashed traces).

Figure 9:

Transient, slight modulation of ongoing-spontaneous spiking activity and ambient GABA concentration by subliminal stimulation. (A) Spiking activity. The open circles denote action potentials in P cells when presented with a subliminal stimulus (f2). The filled circles denote those without stimulation. The ongoing-spontaneous spiking activity is transiently (more than about 500 msec) slightly modulated by the subliminal stimulus (see the dashed squares and ovals). No learning process took place. Inside the dashed squares, the emergence of open circles indicates a small increase in ongoing-spontaneous spiking activity, and their disappearance inside the dashed ovals indicates a small decrease in ongoing-spontaneous spiking activity; see also Figure 5B. (B) Ambient GABA concentrations around stimulus-relevant (see trace n = 2) and stimulus-irrelevant (see trace n = 1 and n = 3) P cells. The subliminal stimulus (f2) was presented (solid traces) or not (dashed traces).

As shown in Figure 9B, the modulation of ongoing-spontaneous neuronal activity is caused by transient changes in local levels of ambient GABA. The subliminal stimulation decreases the ambient GABA concentration around stimulus-relevant P cells (see the solid trace, n = 2), while increasing those around stimulus-irrelevant P cells (see the solid traces, n = 1 and n = 3). Transporters on Ia and Ib cells contribute to these concentration changes (see appendix  B). A fraction of spikes evoked by the subliminal stimulus (see the open circles within the dashed rectangle in Figure 9A) activates their axonal target Ia cells below firing threshold (see the inset). Notably, one or two spikes evoked in a P cell are insufficient to evoke an action potential but sufficient to elicit a subthreshold response in its target Ia cell, that is, to depolarize the Ia cell below firing threshold. The subthreshold depolarization of Ia cells drives their transporters to export GABA into the extracellular space around their axonal target Ib cells, which then hyperpolarizes the Ib cells and drives transporters to import (remove) GABA from the extracellular space around their axonal target P cells (see the solid trace n = 2).

Figure 10A presents the average synaptic weight between P cells belonging to stimulus-relevant cell assembly (n = 2) or stimulus-irrelevant cell assemblies (n = 1 and n = 3) as a function of learning time. We found that the synaptic enhancement (see the circles) is completed within a few seconds after the termination of subliminal stimulation (time = 0.01 sec). Figure 10B shows the dependence of synaptic weight between stimulus-relevant P cells on the number of cell units within cell assemblies (see Table 1): N = 20, 40, 100. The filled circles denote initial weights, and the open circles denote weights after the learning: meanSD. As shown in Figure 10C, the correlation of ongoing-spontaneous activity between stimulus-relevant P cells increases after the learning, which evidences a memory trace for the subliminally presented stimulus.

Figure 10:

Time courses of synaptic alteration. (A) Average synaptic weight between P cells belonging to the stimulus-relevant cell assembly (n = 2; see the circles) or stimulus-irrelevant cell assemblies (n = 2 and n = 3; see the triangles and squares) as a function of learning time. The subliminal stimulus (f2) was presented at time = 0 for 10 msec. (B) Dependence of synaptic weight between stimulus-relevant P cells on the number of cell units within cell assemblies (see Table 1): N = 20, 40, 100. The filled circles denote initial weights employed for successful subliminal learning, and the open circles denote weights after the learning: meanSD. (C) Cross-correlation functions of ongoing-spontaneous membrane potentials after subliminal learning, calculated for five pairs of stimulus-relevant P cells that were arbitrarily chosen. OS-STDP (see the solid traces) or one-shot STDP (see the dashed traces) took place.

Figure 10:

Time courses of synaptic alteration. (A) Average synaptic weight between P cells belonging to the stimulus-relevant cell assembly (n = 2; see the circles) or stimulus-irrelevant cell assemblies (n = 2 and n = 3; see the triangles and squares) as a function of learning time. The subliminal stimulus (f2) was presented at time = 0 for 10 msec. (B) Dependence of synaptic weight between stimulus-relevant P cells on the number of cell units within cell assemblies (see Table 1): N = 20, 40, 100. The filled circles denote initial weights employed for successful subliminal learning, and the open circles denote weights after the learning: meanSD. (C) Cross-correlation functions of ongoing-spontaneous membrane potentials after subliminal learning, calculated for five pairs of stimulus-relevant P cells that were arbitrarily chosen. OS-STDP (see the solid traces) or one-shot STDP (see the dashed traces) took place.

3.3  Learning for Too Weak Stimuli

In the previous sections, we showed how subliminal stimuli that were too brief to perceive could be memorized. As I addressed in section 1, stimuli that are too weak are another type of subliminal stimuli. In this section, we show how they could be memorized.

As shown in Figure 11A, bursts of spikes are evoked in P cells when presented with a supraliminal stimulus (see the left horizontal line). Then we applied a subliminal stimulus (see the center horizontal line) whose time period was long but intensity was too weak to evoke a burst of spikes. During the stimulation period, a learning process took place. After the learning, we applied the same supraliminal stimulus (see the right horizontal line). We found that the learning can enhance the responsiveness of the network to that supraliminal stimulus. The second and bottom traces are ambient GABA concentrations around P and Ib cells belonging to respective cell assemblies (). Note that a decrease in ambient GABA concentration around stimulus-relevant P cells (see the upper arrow) enhances their activity, in favor of which an increase in ambient GABA concentration around stimulus-relevant Ib cells operates (see the lower arrow).

Figure 11:

Learning for a subliminal stimulus whose time period is long but intensity is too weak for the network to respond. (A) Top: Raster plots of action potentials evoked in P cells belonging to respective cell assemblies (). A supraliminal stimulus was presented to the network (f2; see the left horizontal line). When presented with a subliminal stimulus (; see the center horizontal line), a learning process took place, and synaptic weights between P cells were modified. The same supraliminal stimulus was presented to the network after the learning (f2; see the right horizontal line). Middle: Ambient GABA concentrations around P cells. Bottom: Ambient GABA concentrations around Ib cells. (B) Raster plots and ambient GABA concentrations without changing ambient GABA levels during the learning period. (C) Dependence of P cell spiking activity (firing rate), evoked by the second supraliminal stimulus, on input current used for the learning (see in panels A and B) during which the ambient GABA levels were allowed to change (circles) or not (triangles). The arrows indicate the minimal input currents that ensure successful subliminal learning.

Figure 11:

Learning for a subliminal stimulus whose time period is long but intensity is too weak for the network to respond. (A) Top: Raster plots of action potentials evoked in P cells belonging to respective cell assemblies (). A supraliminal stimulus was presented to the network (f2; see the left horizontal line). When presented with a subliminal stimulus (; see the center horizontal line), a learning process took place, and synaptic weights between P cells were modified. The same supraliminal stimulus was presented to the network after the learning (f2; see the right horizontal line). Middle: Ambient GABA concentrations around P cells. Bottom: Ambient GABA concentrations around Ib cells. (B) Raster plots and ambient GABA concentrations without changing ambient GABA levels during the learning period. (C) Dependence of P cell spiking activity (firing rate), evoked by the second supraliminal stimulus, on input current used for the learning (see in panels A and B) during which the ambient GABA levels were allowed to change (circles) or not (triangles). The arrows indicate the minimal input currents that ensure successful subliminal learning.

As shown in Figure 11B, the responsiveness cannot be enhanced if ambient GABA concentrations are not allowed to change during the learning period. Figure 11C presents the dependence of neuronal activity (firing rate), evoked by the second supraliminal stimulus, on input current used for the learning (see in Figures 11A and 11B). The modulation of ambient GABA levels reduces the minimal input current beyond which the learning could be made successfully (see the open arrow). With ambient GABA levels unchanged, larger input current ( 122 pA) is required for successful learning (see the filled arrow). These results indicate that the modulation of local ambient GABA levels contributes to improving the learning of subliminal stimuli that are too weak to perceive.

3.4  Influence of Reciprocal Excitatory-Inhibitory connection

To assess an influence of reciprocal excitatory-inhibitory (E-I) connection on subliminal learning, we made a simulation. Figure 12A shows raster plots for enhanced learning in which the subliminal stimulation period was extended from 10 msec to 20 msec. The original model (see the inset and Figure 1) was employed. Due to a resultant excessive increase in synaptic weight, the burst firing continues even after the termination of the (second) supraliminal stimulus. The prolonged burst firing interferes with preparing for the next sensory input. As shown in Figure 12B (top: a revised model), we can avoid that problem if considering the reciprocal P-Ia (E-I) connection (see the inset); namely, the burst firing ceases soon after the stimulus termination (see the arrow).

Figure 12:

Roles of reciprocal excitatory-inhibitory (E-I) connection. (A) Raster plots for enhanced learning in which the subliminal stimulation period was extended from 10 msec to 20 msec. The original model (see the inset and Figure 1) was employed. The burst firing continues even after the termination of the (second) supraliminal stimulus. (B) Raster plots for a revised model in which the reciprocal P-Ia connection was considered (see the inset). Top: Ongoing-spontaneous spike-timing-dependent plasticity (OS-STDP). The burst firing ceases soon after the stimulus termination (see the arrow). Bottom: One-shot spike-timing-dependent plasticity (one-shot STDP). The learning period was restricted to the stimulus presentation period.

Figure 12:

Roles of reciprocal excitatory-inhibitory (E-I) connection. (A) Raster plots for enhanced learning in which the subliminal stimulation period was extended from 10 msec to 20 msec. The original model (see the inset and Figure 1) was employed. The burst firing continues even after the termination of the (second) supraliminal stimulus. (B) Raster plots for a revised model in which the reciprocal P-Ia connection was considered (see the inset). Top: Ongoing-spontaneous spike-timing-dependent plasticity (OS-STDP). The burst firing ceases soon after the stimulus termination (see the arrow). Bottom: One-shot spike-timing-dependent plasticity (one-shot STDP). The learning period was restricted to the stimulus presentation period.

We evaluated the response property of the revised model, in which the OS-STDP (see Figure 12B: top) or one-shot STDP (Figure 12B: bottom) scheme was employed. It was found that the enhancement of the responsiveness to the second supraliminal stimulus is greater through OS-STDP than through one-shot STDP. This leads to the same conclusion reached by the original model: ongoing-spontaneous synaptic alteration through STDP following subliminal stimulation contributes to leaving memory traces for subliminal sensory stimuli in cortical circuitry (see Figure 4). This simulation result indicates that the proposed (original) network model is robust to the addition of the Ia-to-P connection (see the inset of Figure 12B).

4  Discussion

Perception of supraliminal stimuli in general might be reflected in bursts of action potentials (spikes), and their memory traces could be formed through spike-timing-dependent plasticity (STDP). Memory traces for subliminal stimuli might be formed in a different manner because subliminal stimulation evokes a fraction (but not a burst) of spikes. We showed that a subliminal stimulus that was too brief (10 msec) to perceive transiently (more than about 500 msec) depolarized stimulus-relevant principal cells and hyperpolarized stimulus-irrelevant principal cells in a subthreshold manner (see Figure 8). This led to a small increase or decrease in ongoing-spontaneous spiking activity frequency (less than 1 Hz on average; see Figures 5 and 9). Synaptic modification based on STDP during this time period effectively enhanced relevant synaptic weights, by which subliminal learning was improved. GABA transporters on GABAergic interneurons modulated local levels of ambient GABA. Ambient GABA molecules acted on extrasynaptic receptors, provided principal cells with tonic inhibitory currents, and contributed to achieving the subthreshold neuronal state. We suggest that ongoing-spontaneous synaptic alteration through STDP following subliminal stimulation may be a possible neuronal mechanism for leaving its memory trace in cortical circuitry. Regulation of local ambient GABA levels via transporter-mediated GABA import and export may be crucial for subliminal learning.

To the best of our knowledge, OS-STDP is not experimentally evident. However, we consider that synaptic alteration is unlikely to be restricted solely to sensory stimulation periods. Rather, it may take place in various behavioral situations, including an ongoing-spontaneous neuronal state that would be unconsciously influenced by subliminal sensory stimulation. We found that compared to one-shot STDP, OS-STDP is sensitive to background firing. This is because OS-STDP requires occasional synchronization of firings during the ongoing-spontaneous activity time period following subliminal stimulation.

We showed that subliminal learning decreased the ambient GABA concentration around stimulus-relevant principal cells, while increasing those around stimulus-irrelevant principal cells (see Figure 7B). This result indicates that synaptic modification by learning entails alteration of ambient GABA levels. An experiment by Massie and colleagues (2003) may in part support this notion. After cortical reorganization following visual deprivation in cat visual cortex, an increase in ambient (extracellular) GABA concentration occurred in nondeprived cortical areas. The stimulus-irrelevant cell assemblies, in which ambient GABA concentrations were increased (see the solid traces, n = 1 and n = 3, in Figure 7B), may correspond to such nondeprived cortical areas. Contrary to their expectation (Arckens et al., 2000), they failed to detect a decrease in ambient GABA concentration in the deprived cortical area, which may correspond to a decrease in ambient GABA concentration in the stimulus-relevant cell assembly (see the solid trace, n = 2, in Figure 7B).

We showed that a decrease in ambient GABA concentration around stimulus-relevant principal cells (see the solid trace, n = 2, in Figure 9B) enhanced synaptic plasticity. A recent experiment (Maya-Vetencourt et al., 2012) showed a similar result. In the visual system, the researchers demonstrated that treatment of insulin-like growth factor 1 (IGF-1) restored the susceptibility of cortical neurons to monocular deprivation and promoted the recovery of normal visual function in adult amblyopic animals, indicating the enhancement of adult visual cortex plasticity. Their notable finding was that the synaptic enhancement was accompanied by a marked reduction of ambient GABA concentration in the cortex. They concluded that reducing local ambient GABA levels could enhance visual cortical plasticity. Our simulation result indicated the importance of regulating local ambient GABA levels for the enhancement of synaptic plasticity in subliminal learning.

Ress and Heeger (2003) made a functional magnetic imaging (fMRI) study and measured activity in human early visual cortical areas (V1, V2, V3) during a perceptual task: detection of visual gratings (bars of particular orientations) on a background (noise or plaid) pattern. Behavioral responses were recorded and grouped into four perceptual categories: hit, false alarm, miss, and correct reject. Hit is a correct response to a stimulus, false alarm an incorrect response to the absent stimulus, miss a missed response to the stimulus, and correct reject a correct response to the absent stimulus. Significant cortical activity was observed in hits (and false alarms) but not in misses (and correct rejects).

As is well known, fMRI measurements are based on blood oxygenation level-dependent (BOLD) signals. A recent experimental study (Kahn et al., 2013) indicated a strong correlation between spike rate and BOLD signal. Heeger and Ress (2002) indicated that single-unit firing rates were roughly proportional to BOLD signals. Based on these studies, we assumed that the perception of supraliminal stimuli (as hits) might induce high firing activity (i.e., bursting; see the top panel of Figure 5A) reflected in high BOLD signal, and the missing perception of subliminal stimuli (as misses) low firing activity (i.e., sparse spiking but not bursting; see the middle panel of Figure 5A) reflected in low BOLD signal.

The cortex is known to have diverse types of GABAergic interneurons, including basket and chandelier cells (Gupta, Wang, & Markram, 2000). The researchers made multiple neuron recordings and revealed a large number of interneuron connections. It was suggested that such GABAergic interneuron diversity might provide combinatorial inhibitory effects in the cortex. In the neural network model presented here, the hypothetical projection between GABAergic interneurons, Ia-to-Ib, contributed to regulating excitability in P cells.

Researchers (Sillito, 1984; Eysel, 1992) have suggested that lateral inhibition between principal cells, which is mediated by interneurons, plays important roles in neuronal information processing, such as tuning to sound frequency (Yost, 1994) in the primary auditory cortex, to orientation or direction (Eysel, Shevelev, Lazareva, & Sharaev, 1998), and to cross-orientation inhibition (Kisvarday, Kim, Eysel, & Bonhoeffer, 1994) in the primary visual cortex. It is well known in primary visual cortex (Stemmler, Usher, & Kock, 2000) that pyramidal cells directly excite nearby cells within the same orientation column and inhibit indirectly cells in neighboring columns with different orientation preferences. Based on these studies, we assumed direct connections between principal cells within the same cell assembly and indirect connections to principal cells in different cell assemblies via Ib cells in order to exert lateral inhibition.

Appendix A:  The Neural Network Model

Dynamic evolution of membrane potential of the ith P cell belonging to cell assembly n is defined by
formula
A.1
where is an excitatory (synaptic) current from other P cells, an inhibitory (synaptic) current from an Ib cell, an ambient GABA-mediated inhibitory (nonsynaptic) current via extrasynaptic receptors, and an input current when the network is presented with a sensory feature stimulus . These currents are defined by
formula
A.2
formula
A.3
formula
A.4
formula
A.5
Dynamic evolution of the membrane potential of the ith Ia cell is defined by
formula
A.6
where is an excitatory current from a P cell and is an ambient GABA-mediated inhibitory current. These currents are defined by
formula
A.7
formula
A.8
Dynamic evolution of the membrane potential of the ith Ib cell is defined by
formula
A.9
where and are excitatory and inhibitory currents from P and Ia cells, respectively. is an ambient GABA-mediated inhibitory current. These currents are defined by
formula
A.10
formula
A.11
formula
A.12

In these equations, is the fraction of AMPA receptors in the open state triggered by presynaptic action potentials of the jth P cell belonging to cell assembly n at time t. and are the fractions of intrasynaptic GABAa receptors in the open state triggered by presynaptic action potentials of the ith Ib and Ia cells, respectively. , where Y = P, Ia or Ib, is the fraction of extrasynaptic GABAa receptors, located on the ith Ycell, in the open state provoked by ambient GABA. (For the details of model parameters and their values, see Table 1.)

Appendix B:  Dynamics of Receptor and Transporter

Receptor dynamics is defined by
formula
B.1
formula
B.2
formula
B.3
where and are concentrations of glutamate and GABA in the synaptic cleft, respectively. = and = for 1 msec when the presynaptic jth P cell and type X cell fire, respectively. Otherwise, = 0 and = 0.
Dynamics of ambient GABA concentrations around P and Ib cells, to which Ib and Ia cells project (see Figure 1A), is defined by
formula
B.4
formula
B.5
Concentrations of ambient GABA around Ia cells are simply set to , because the Ia cells do not receive GABAergic innervations. For the details of model parameters and their values, see Table 1 and our previous studies (Hoshino, 2009, 2010, 2012, 2013a, 2013b, 2014).

Appendix C:  Action Potential Generation

The probability of firing of the jth Y cell belonging to cell assembly n is defined by
formula
C.1
When a cell fires, its membrane potential is depolarized toward mV, which is kept for 1 msec and then reset to the resting potential. For the details of model parameters and their values, see Table 1 and our previous studies (Hoshino, 2007a, 2007b, 2008, 2011a).

Appendix D:  Synaptic Modification

The synaptic connection weight between the presynaptic jth P cell and the postsynaptic ith P cell (see in equation A.2) is modified according to the relative timing between pre- and postsynaptic action potentials (Bi & Poo, 1998; Young et al., 2007), which is defined by
formula
D.1
formula
D.2
In these equations, is a time difference between presynaptic (tpre) and postsynaptic (tpost) action potentials. For the details of model parameters and their values, see Table 1 and Hoshino (2011b).

Acknowledgments

I express my gratitude to Takeshi Kambara for his helpful discussions and to reviewers for giving me valuable comments and suggestions.

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