Cerebral Cortex Advance Access originally published online on October 9, 2006
Cerebral Cortex 2007 17(8):1766-1781; doi:10.1093/cercor/bhl088
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First-In–First-Out Item Replacement in a Model of Short-Term Memory Based on Persistent Spiking
Center for Memory and Brain, Department of Psychology and Program in Neuroscience, Boston University, Boston, MA 02215, USA
Address correspondence to R.A. Koene, Center for Memory and Brain, Department of Psychology and Program in Neuroscience, Boston University, 64 Cummington Street, Boston, MA 02215, USA. Email: randalk{at}bu.edu.
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Abstract |
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Persistent neuronal firing has been modeled in relation to observed brain rhythms, especially to theta oscillations recorded in behaving animals. Models of short-term memory that are based on such persistent firing properties of specific neurons can meet the requirements of spike-timing–dependent potentiation of synaptic strengths during the encoding of a temporal sequence of spike patterns. We show that such a spiking buffer can be simulated with integrate-and-fire neurons that include a leak current even when different numbers of spikes represent successive items. We propose a mechanism that successfully replaces items in the buffer in first-in–first-out (FIFO) order when the distribution of spike density in a theta cycle is asymmetric, as found in experimental data. We predict effects on the function and capacity of the buffer model caused by changes in modeled theta cycle duration, the timing of input to the buffer, the strength of recurrent inhibition, and the strength and timing of after-hyperpolarization and after-depolarization (ADP). Shifts of input timing or changes in ADP parameters can enable the reverse-order buffering of items, with FIFO replacement in a full buffer. As noise increases, the simulated buffer provides robust output that may underlie episodic encoding.
Key Words: gamma rhythm • integrate-and-fire neurons • persistent firing • sequence buffer • short-term memory • theta rhythm
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Introduction |
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Computer simulations of the encoding of a temporal sequence of spike patterns in recurrent neuronal networks often rely on a Hebbian model (Hebb 1949
) of spike-timing–dependent potentiation (STDP) for the update of synaptic strengths. During learning, these simulations depend on the repeated presentations of spike patterns that are separated by time intervals smaller than about 40 ms, a window within which consecutive pre- and postsynaptic spikes were found to elicit long-term potentiation (Levy and Stewart 1983; Markram et al. 1997; Bi and Poo 1998). Yet in realistic experiments, a subject may be exposed to a sequence of stimuli that are separated by greater time intervals. One way to deal with this problem is to store each sequence in a short-term buffer that presents items at appropriate intervals. Because such a buffer cannot depend on synaptic modification for the temporary storage, we presuppose a mechanism of persistent spiking. The mechanism relies on after-depolarization (ADP), a membrane response that is intrinsic to many pyramidal cells of the entorhinal cortex (EC) (Klink and Alonso 1997a, 1997b; Egorov et al. 2002).
Computer models by Lisman and Idiart (1995) and Jensen et al. (1996) have simulated short-term memory (STM) function based on persistent spiking during modulation by theta rhythm. Chrobak and Buzsáki (1998) obtained elecrophysiological data during task performance that demonstrated nested oscillations at theta (8–9 Hz) and gamma (40–100 Hz) frequencies in EC, as well as similar activity in dentate, hilar, and hippocampal regions. These data also showed a phase-shifted synchronization of the oscillations in different regions. Previous simulations based on the Lisman and Idiart model did not deal with the dynamic characteristics of leaky spiking neurons and provided no mechanism for the ordered replacement of items in a buffer that is filled to capacity. Four-item capacity and replacement of items in a first-in–first-out (FIFO) manner are a good fit to the recency portion of graphs of serial position data (Atkinson and Shiffrin 1968; Kahana 1996). Psychophysical evidence for ordered item displacement has been gathered in tests ranging from precategorical acoustic storage (Crowder and Morton 1969) to observed interaction between memory load and item position for semantic information (Haarmann and Usher 2001). Earlier models incorporate functions that specifically attempt to fit this evidence (Phillips et al. 1967; Kahana 1996).
In prior work, we proposed buffer models that are related to the STM models by Lisman and Idiart (1995) that are based on persistent spiking, but we include more detailed simulation of the asymmetric distribution of spiking activity in each cycle of the 3- to 12-Hz theta brain rhythm (Koene 2006a, 2006b, in preparation), and we use leaky integrate-and-fire neurons to simulate neural dynamics (Koene et al. 2003). The implementation of the more realistic asymmetry has functional benefits: distinct portions of each theta cycle in a buffer can be synchronized with distinct modes of synaptic encoding and retrieval that alternate in each cycle of a theta rhythm in the recurrent networks that receive spike input from that buffer (Koene et al. 2003; Koene and Hasselmo 2005). By contrast, Jensen et al. (1996) relied on temporally separated periods of at least several seconds that are devoted exclusively to associative encoding or to retrieval in order to avoid interference between spiking in those 2 modes. Here, we add to the model a plausible mechanism for the FIFO ordered replacement of items that are maintained in the buffer as successive patterns of simultaneous spikes. Although Jensen et al. (1996) described a method by which items in the buffer could be replaced as new input appears, that proposal was not evaluated in simulations, and their proposed method relied on an even distribution of item reactivation spikes throughout a theta cycle. The mechanism proposed here performs the ordered replacement of buffered items, even when different numbers of simultaneous spikes represent each item, as may be expected for realistic stimuli. Our previously published studies involved the extensive use of an integrate-and-fire version of the STM buffer model in which pattern reactivation occurs specifically at the depolarized phase of theta modulation. Those studies focused on hippocampal (Hasselmo, Cannon, and Koene 2002; Koene et al. 2003) and prefrontal cortex functions (Koene and Hasselmo 2005; McGaughy et al. 2005) during behavioral tasks in which the results of neuronal computation elicited dynamic responses from the task environment but did not deal specifically with the issue of patterns with different numbers of simultaneous spikes that represent the items maintained in STM.
Different numbers of neurons may spike to represent 1) the last item reactivated in the ordered buffer, 2) the insertion of a new item into the buffer by input, and 3) the first item that may need to be removed from the buffer. A mechanism of item replacement that is agnostic about numbers of spikes can use theta phase information to achieve the required functions. The necessary phase information is shown in Figure 1 for a theoretical 2-item buffer. Item replacement depends on the occurrence of spikes in both of 2 phases of the buffer cycle, namely, the last item phase indicated by (a) in Figure 1 and the input phase indicated by (b) in Figure 1. If this condition is met, then the item replacement mechanism must suppress spiking of any number of buffer neurons at the phase of the cycle indicated by (c) in Figure 1 at which spiking of a buffered first item would be reactivated.
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In the following sections, we describe the STM buffer model, as implemented in the Catacomb (Cannon et al. 2003
) simulation environment and its performance with different numbers of spikes representing individual items. The mechanism postulated for FIFO replacement of buffered items is then presented. The functional effects of significant changes of specific parameters of the STM model are analyzed, and modifications that lead to FIFO buffering in the reversed order of item presentation are demonstrated. Finally, the buffer model is shown to exhibit sufficient robustness in the presence of increasing levels of noise that affect spike timing to act as a useful STM when novel items need to be sustained for encoding in episodic memory. |
Methods |
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Our model is based on data suggesting that many pyramidal cells in layer II of entorhinal cortex (ECII) are specifically suited to reactivate firing patterns in a persistent manner due to intrinsic neuronal mechanisms. Such neurons exhibit an ADP of membrane potential after an action potential is elicited at the membrane. Klink and Alonso (1997a)
showed that the ADP is caused by calcium-sensitive cation currents that are induced by muscarinic receptor activation. In addition to the ADP, activity in EC is modulated by a strong brain rhythm known as the theta rhythm that is present in behaving animals (Buzsáki et al. 1983; Fox et al. 1986; Fox 1989; Kamondi et al. 1998). In rats, for example, the theta rhythm has a frequency of about 6–10 Hz, which we represent in most simulations with 8 Hz. This rhythm may synchronize activity within a STM based on persistent spiking (Koene et al. 2003), and it may synchronize output to other regions such as region CA3 (Hasselmo, Bodelon, and Wyble 2002) or ECIII (Koene et al. 2003), where corresponding synaptic changes may occur. The reactivation of firing patterns by ADP in an ECII population of neurons between specific phases of each cycle of theta rhythm can therefore provide reliable input to other neuronal populations. Our model of STM in EC builds on a model first proposed by Lisman and Idiart (Lisman and Idiart 1995; Jensen and Lisman 1996a). In both models, recurrent inhibition separates the reactivation of sequential spike patterns in the buffer. The inhibition is provided by a population of interneurons, all of which spike in response to an action potential at any pyramidal cell in ECII (Bragin et al. 1995). A Leaky Integrate -and -fire Neuron Implementation of the STM Buffer
Equation (1) shows how various contributions affect the response of model ECII pyramidal neurons and therefore their ability to sustain ordered sequences of spikes.
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v is the change of the membrane potential V during a small time interval t, and gi is the conductance and Erev,i the reversal potential of a contributing membrane current. Capacitance and leak current determine the characteristic response of the membrane potential as it gradually returns to the resting potential. Below, we describe the contributing currents provided by rhythmic theta modulation, external afferent input to the buffer cells, intrinsic after-hyperpolarizing and after-depolarizing responses to action potentials, and inhibitory synaptic input from recurrent fibers of an interneuron network that is responsible for the observed gamma rhythm.
We use integrate-and-fire simulations of pyramidal cells of ECII, which have a membrane response with exponential decay to a resting potential Erest = –60 mV due to membrane capacitance of C = 1 mF and time constant of the leak conductance 
leak = 9 ms (the STM buffer has been shown to work with time leak constants between 8 and 19 ms. Below 8 ms, spike reactivation does not occur, whereas spikes may reactivate twice in one theta period when the time constant is above 19 ms, which leads to changes in the order of the spike patterns that are sustained in the buffer). The current contributed to the membrane potential by the leak conductance is defined by Erest and the conductance gleak = C/leak. The firing threshold is –50 mV, and the resting potential of –60 mV is chosen as the reset potential that follows a spike. Action potentials have a duration of 1 ms and are followed by a 2-ms refractory period and subsequent strong after-hyperpolarization (AHP). Many currents in equation (2), such as the AHP and synaptic input, are simulated with a fixed reversal potential Erev,i and a biexponential conductance response function
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fall,i and rise,i are its fall and rise time constants, and anorm is a normalizing factor to insure that the maximum value of gi(t) is Gi. The normalizing factor is computed by
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The AHP response approximates a single exponential function through a small rise time constant rise,AHP = 10–4 ms and an exponential decay time constant fall,AHP = 30 ms with amplitude GAHP = 23 ns and has a reversal potential of EAHP = –90 mV. The biexponential ADP response has the shape of an alpha function with time constants rise,ADP = fall,ADP = 125 ms, amplitude GADP = 30 ns, and the reversal potential EADP = –45 mV. The ADP must rise over a time span that is approximately equal to the period of a theta cycle because reactivation of spikes on the rising flank of the ADP response insures that the order of spiking is maintained.
We model the rhythmic oscillation of the membrane potentials of pyramidal cells in ECII as inhibitory synaptic input that is modulated by rhythmic activity at a frequency of 8 Hz originating in the medial septum (Alonso et al. 1987; Stewart and Fox 1990; Skaggs et al. 1996; Wallenstein and Hasselmo 1997; Brazhnik and Fox 1999). The biexponential synaptic responses that cause this modulation have reversal potential Etheta = –90 mV, a conductance amplitude of Gtheta = 10 ns, and time constants rise,theta = 0.1 ms and fall,theta = 20 ms. This gives the modulation of membrane potential the typical asymmetrical shape of the theta rhythm. The simulated 8-Hz theta rhythm is driven by septal spikes that commence at t = 0 ms.
An interneuron population in ECII is simulated by a single interneuron that receives fast connections (fast connections are simulated with a transmission delay of between 0 and 1 ms) from all pyramidal buffer cells and provides recurrent input to all pyramidal buffer cells. The interneuron responds rapidly to any buffer output spikes, due to biexponential synaptic input with the characteristic parameters G = 30 ns, rise = 1 ms, and fall = 2 ms, and a postsynaptic reversal potential Erev = 0 mV. This "gamma" interneuron recuperates quickly due to a fast AHP response with EAHP = –90 mV, GAHP = 100 ns, rise,AHP = 10–4 ms, and fall,AHP = 4 ms. The interneuron model also uses equations (1–5) but differs from that of the pyramidal cells due to reset and rest potentials Ereset = Erest = –70 mV and a leak time constant leak = 10 ms. The competitive inhibition provided to pyramidal buffer cells by the model interneuron population simulates the postsynaptic response of 
-aminobutyric acid (GABAA) receptors as a membrane current with Egamma = –70 mV and a biexponential conductance response with characteristic parameters Ggamma = 100 ns, rise,gamma = 0.1 ms, and fall,gamma = 2.5 ms. The model of the network of interneurons in ECII was investigated in prior (Koene 2001) and related (Koene 2006b, in preparation) work.
The transmission of new input spikes to the buffer as well as the transmission of recurrent inhibition that is observed as a power variation in the gamma frequency band of the electroencephalography are modulated at theta frequency due to GABAB receptor activation by septal input at presynaptic terminals. In our simulations, the dynamic shape of the modulating amplitude, ftmod(t), is generated by the normalized output of a model neuronal membrane that receives synaptic input at theta frequency. The resulting transmission modulation has a scalloped shape typical of observed theta modulation. The transmission modulated conductance responses of input (ginput(t)ftmod,input(t)) and of recurrent inhibition (ggamma(t)ftmod,gamma(t)) are 180 degrees out of phase. This means that input is suppressed during the reactivation phase of the buffer, at which time recurrent inhibition is needed to maintain the separation of successive spike patterns. Conversely, recurrent inhibition is suppressed when new input is received at the input phase of the buffer (which may initiate item replacement if the buffer is full). The period of greatest depolarization due to membrane currents produced by the theta rhythm (with Etheta and gtheta(t)) is in phase with the strongest recurrent inhibitory transmission from the gamma interneuron population.
As in Lisman and Idiart (1995), spikes produced in ECII by stimulus input are reactivated repeatedly by the combination of ADP and positive theta modulation of membrane potential at a rate that matches the frequency of theta oscillation. A pattern of spikes that represents a stimulus input is maintained by this persistent spiking without any prerequisite synaptic connectivity. The persistent reactivation of a spike due to ADP is shown in Figure 2. The effect of ADP is modulated by acetylcholine (ACh), low levels of which do not support persistent firing (low ACh in Fig. 2). At high levels of ACh (high ACh in Fig. 2), stimulated buffer neurons can sustain firing at a rate that is determined by the time course of the ADP response. When ADP is combined with the alternating hyperpolarizing and depolarizing modulation of buffer neurons by the theta brain rhythm (high ACh + theta in Fig. 2), then persistent reactivation is phase locked within the cycle frequency of the theta rhythm. Rhythmic septal spikes that initiate each cycle of the membrane oscillation and the modulation of afferent and recurrent transmission have a 112-ms offset, whereas the first afferent can appear at t = 125 ms (i.e., input appears with a 13-ms relative phase offset after the peak of theta depolarization). The phase offset of all rhythmic input or modulation in the following paragraphs is given as an offset from t = 0 ms.
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The membrane potentials of 3 neurons of a STM buffer are plotted in Figure 3. Theta oscillations define 2 functional phases of the buffer neurons. We call the phase interval of greatest rhythmic depolarization the reactivation phase of STM and the remaining interval the input phase of STM. The plots show that spiking produced by afferent activity during the input phase of the buffer is reactivated by the ADP during subsequent reactivation phases. The duration of the rise of ADP matches the period of oscillation. This means that the ADP of the earliest neuron to spike in one cycle allows that neuron to reach threshold first in the following cycle. The order of spikes is maintained during reactivation in STM. As spikes caused by output of the buffer occur in pre- and postsynaptic neurons of a target population with recurrent fibers such as entorhinal layer III or region CA3 of the hippocampus, an asymmetric function of STDP can take into account the order of spikes. This ensures that long term potentiation (LTP) (Bliss and Lømo 1973
; Bliss and Collingridge 1993) is elicited in specific connections so that the order of events is maintained in episodic memory.
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A rapid succession of septal spikes is used to suppress action potentials in the buffer neurons between trials of an experiment. This simulates the disappearance or resetting of theta rhythm as observed during context switches (Wyble et al. 2004
). The buffer is cleared for the next trial. A Phase-Locked Mechanism of FIFO Replacement of Buffered Items
In the absence of input, the contents of a STM buffer decay gradually due to noise and a slow AHP (modeled as a biexponential response with Erev = –70 mV, G = 0.01 ns, rise = fall = 3000 ms, an alpha function). When a buffer that is filled to capacity receives new input at a rapid pace, so that the loss of spikes neither due to effects of noise nor due to slow AHP remove items from the buffer before the new input arrives, continued ordered buffering of the most recent spike patterns is only possible if a mechanism exists that can selectively remove the representation of the first item from the buffer to make room for a representation of the new item. If each item can be represented by a different number of simultaneous spiking pyramidal cells in the buffer, then the mechanism must be able to remove the first item regardless of how many and which neurons spiked to represent that item, and it must do this only when the buffer is full and new input arrives (if the buffer state is not detected and item replacement is elicited for every afferent input, then the effective buffer has the capacity to maintain one item). Neither the arrival of new input nor the "full" state of the buffer can be determined simply by detecting the activity of a certain number of spikes within a given time interval because the representations of the afferent input and of the reactivated items in the buffer may each consist of any number of spikes. In a buffer that depends on the timing of input and of gamma subcycles within each theta cycle, a reliable indicator of full buffer state and of an input event is the presence of spikes at specific phases of the theta cycle. In particular, spikes at the input phase of the theta cycle indicate the arrival of new afferent input and spikes at the reactivation phase of the Nth item in a buffer with a capacity of N items indicate that the buffer is filled to capacity.
We therefore propose a FIFO ordered item replacement mechanism for the STM buffer that is robust with regard to differences in the number of spikes that represent specific items in the buffer. Our model for the replacement mechanism assumes that in addition to cells with ADP (PADP in Fig. 4), ECII contains at least 3 other functionally distinct populations of neurons (schematically drawn as single nodes in Fig. 4, all with Erest = Ereset = –60 mV). These are 2 populations of pyramidal cells (Pf and Pi in Fig. 4, both with leak = 9 ms and biexponential fast AHP responses with Erev = –90 mV, G = 10 ns, rise = 0.1 ms, and fall = 50 ms) that do not exhibit ADP and one population of interneurons (Ir in Fig. 4, with leak = 10 ms) that is not already recruited to participate in the recurrent gamma inhibition.
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Pyramidal cells of the Pf population receive as input (with biexponential postsynaptic responses at each synapse specified by Erev = 0 mV, G = 6 ns,
rise = 0.1 ms, and fall = 1 ms) the spikes produced by the pyramidal buffer cells (the PADP trace in Fig. 4). This input is transmission modulated by activity on GABAB receptors on the terminals of the input synapses, so that the input gated in this manner is sensitive to the specific phase interval at which the reactivation of the Nth item in the buffer normally occurs. The membrane potential of pyramidal cells in Pf is also modulated by theta rhythm (due to inhibitory biexponential responses with Erev = –90 mV, G = 10 ns, rise = 0.1 ms, and fall = 20 ms) at the same phase as the modulation of the pyramidal cells that provide persistent spiking in the buffer. The synaptic strength of buffer input to the Pf population is tuned to cause spikes in Pf when one or more postsynaptic potentials are elicited, whereas the AHP of the Pf neurons prevents bursting when many simultaneous inputs are received. The Pf population functions as a full buffer state detector that is independent of the number of spikes that are reactivated during each theta cycle of the buffer, as demonstrated by the membrane response of Pf shown in Figure 4. The capacity of the buffer is constrained by the phase offset of the transmission modulation that determines the Nth item reactivation phase at which the Pf population detects a full buffer. In our implementation with an 8-Hz theta rhythm, the buffer is able to maintain up to 5 items (with some restrictions on the input protocol, as explained in the Discussion), and this maximum capacity is specified by a phase offset of 103 ms. A phase offset of 84 ms limits the capacity to 4 items, which is a robust capacity for this buffer. With a phase offset of 68 ms, the buffer capacity is limited to 3 items (in order to allow for some variability of spike timing, which includes the possibility of earlier spiking, the phase offsets were chosen to be 4 ms before the greatest offset at which a full buffer is consistently detected when the desired capacity is reached).
Pyramidal cells of the Pi population receive as input (with Erev = 0 mV, G = 6 ns, rise = 0.1 ms, and fall = 1 ms) the spikes that provide afferent input to the buffer. Pi neurons also receive rhythmic excitatory input at theta frequency (with Erev = 0 mV, G = 2 ns, rise = 0.1 ms, and fall = 20 ms) that is synchronized with the theta modulation of buffer pyramidal cells. This depolarization and the strength of synaptic input due to afferent buffer input spikes are tuned so that Pi neurons spike when one or more afferent input spikes are detected. The AHP of Pi neurons prevents bursting when the input consists of multiple simultaneous spikes. The Pi population acts as an input detector that is independent of the number of spikes that represents the input, as shown in the Pi membrane response trace in Figure 4.
The interneuron population Ir also receives excitatory input (with a biexponential postsynaptic response with Erev = 0 mV, G = 1.2 ns, rise = 0.1 ms, and fall = 10 ms) at theta frequency, at a phase offset (32 ms) that depolarizes the interneurons in synchrony with the expected reactivation phase of the first item that is maintained in the buffer. Input of spikes generated by the Pf pyramidal population is received at synapses with slow biexponential timing constants of the postsynaptic response (Erev = 0 mV, G = 0.5 ns, rise = 20 ms, and fall = 60 ms). Therefore, depolarization by that postsynaptic potential can combine with input of spikes from the Pi pyramidal cells at synapses with faster biexponential response timing (Erev = 0 mV, G = 0.5 ns, rise = 10 ms, and fall = 60 ms) to elicit spiking in the population of interneurons (Ir membrane response trace in Fig. 4). When input indicates both a buffer filled to capacity and the arrival of afferent input, spiking of the Ir interneurons (limited by an AHP with Erev = –90 mV, G = 4 ns, rise = 4 ms, and fall = 50 ms) provides GABAergic inhibitory input to all intrinsically spiking buffer neurons (PADP in Fig. 4) through synapses with reversal potential Erev,Ir = –90 mV, conductance amplitude GIr = 40 ns, and biexponential time constants rise,Ir = 1 ms and fall,Ir = 5 ms. This replacement inhibition specifically suppresses spikes at the phase interval during which the ADP of the spikes that represent the first item maintained in the buffer could elicit reactivation spikes. As the ADP begins to decline, spiking is no longer possible and the item is removed from the buffer. The buffer can then support the maintenance of the new item, as ADP reactivates the spikes elicited by afferent input as the last pattern of spikes in the buffered sequence.
In order to achieve complete first item suppression through a short replacement inhibition at the first item reactivation phase, specific conditions must be met: 1) At first item reactivation time, tfr, the contributions of theta plus ADP begin to elevate membrane potential above the firing threshold. 2) At times t 
tfr, the combined contributions of theta modulation plus ADP plus gamma inhibition always lead to membrane potential below the firing threshold. These 2 conditions are met if the time interval between the previous reactivation time of the first item and tfr is approximately equal to the time taken to reach peak ADP and if theta modulation is flat or its contribution increases more slowly than the contribution of ADP decreases after tfr (if the ADP response exhibits a steeper decline, then a greater proportion of each theta cycle may be used for item reactivation. A more rapid termination of the effect of an intrinsic response [such as the ADP] may be achieved by providing an opposing current with an onset at the desired time of termination [such as an inhibitory postsynaptic potential]).
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Results |
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A buffer mechanism for sequences of spike patterns that is independent of synaptic connectivity is necessary for their encoding in episodic memory. In a realistic setting, the spike patterns that make up a sequence to be encoded in episodic memory may be elicited by input stimuli with intervening time intervals of arbitrary duration. The Hebbian process of STDP depends on pre- and postsynaptic spiking that occurs within a time interval of at most 20–40 ms (Levy and Stewart 1983
; Markram et al. 1997; Bi and Poo 1998). Therefore, a buffer that repeatedly reactivates a time-compressed version of a sequence of spike patterns is ideally suited to meet the requirements for encoding in episodic memory.
Initial simulations illustrated problems with the absence of an item replacement mechanism. Figure 5 shows how distinct input and reactivation modes appear in the persistent spiking buffer and that FIFO replacement is not accomplished simply by adding new input to a full buffer, as was suggested for item replacement in Jensen et al. (1996).
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Testing Buffer Function and the Replacement Mechanism
A simulation of buffer performance with the proposed FIFO item replacement mechanism is shown in Figure 6. The spiking representations of successively presented items (A–F) were acquired in the buffer, and STM was maintained by persistent firing, without synaptic modification. Input spikes (see the afferent spike for item E in Fig. 6c) were presented only once, and item representations consisted of different numbers of simultaneous spikes. Figure 6a shows that representations with 2–8 spikes were maintained in order by a buffer that used a single interneuron model to achieve competitive inhibition. The inhibition appears as a visible gamma oscillation superimposed on the theta rhythm in Figure 6c, during the reactivation of item representations. The sequence of reactivation maintained the order but compressed the time between successive item presentations.
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Figure 6c indicates that the timing of spiking in the model would allow STDP to modify synapses between neurons that are activated by the output of the buffer in order to achieve autoassociative encoding of item representations. The buffer output may also enable episodic encoding with greater STDP time windows (Koene et al. 2003
; Koene and Hasselmo 2005), in which synaptic strengthening by STDP depends on the size of the time interval between spikes (Jensen and Lisman 1996b; Koene 2006a, in preparation). Figure 6b,c shows the spiking of replacement interneurons (I) and the consequent suppression of the first set of persistent spikes. Replacement inhibition removed the representation of item A as item E appeared and removed the representation of item B as item F appeared, so that a buffer capacity of 4 spiking item representations was achieved. In each case, the spike representation of the new item reactivated to take the last position in the buffered sequence. The Significance of Characteristic Model Parameters
We may predict functional changes that result from modifications of specific characteristic parameter values of the buffer model. Transmission modulation of the input to pyramidal cells in the full buffer detector (Pf in Fig. 4) is a critical variable of the proposed mechanism. We predict that such transmission modulation exists in ECII. If it does not, then the alternative approach is to supply phase-specific filtering through strong theta modulation of the pyramidal cells in the full detector. This modulation can take the form of rhythmic excitatory and/or inhibitory input.
Transmission modulation of the recurrent inhibition that is observed as gamma rhythm is not critical, as it merely limits how close together afferent input and the first item reactivation can appear. Without mode-specific modulation, the recurrent gamma inhibition that occurs as a result of pyramidal spikes elicited by afferent input imply that a minimum interval must lie between the afferent input spikes and the first item reactivation spikes (Fig. 7). In the special case of a type of reverse-order STM buffer that depends on the phase of afferent input (demonstrated in Fig. 8), proposed transmission modulation of recurrent inhibition may affect buffer function.
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The buffer works with AHP conductance values between 25 and 35 ns. If the AHP is too strong, it suppresses reactivation (especially reactivation in the same theta cycle, as is necessary when new input is received in the STM buffer). If the AHP is too weak, then different combinations of ADP and theta modulation are needed for a functioning buffer, otherwise a spike pattern may appear more than once in a theta cycle, which causes a deterioration of the order and distinctness of items maintained in the buffer. Changing the conductance or the fall time of the AHP can both change the effective strength of the AHP, with consequences as described above.
An ADP rise time of 90–160 ms results in a working buffer. Our evaluation of the buffer model indicates that there is some room for variations of the intrinsic characteristics. The buffer model does not require extreme fine-tuning of physiological parameters, so that neurons with a range of acceptable characteristic parameter values can participate as buffer neurons in ECII. Buffer function requires precision for phase locking in the following parameters: 1) The phase at which input is received by the buffer. 2) The phase at which a full buffer is detected. 3) The phase at which replacement inhibition occurs in the buffer. Fortunately, phase-specific sensitivity can be learned.
Reverse Buffering
The buffer reactivated and maintained a sequence of spikes in the reverse order of their initial presentation when a specific relationship of characteristic parameters was used, as shown in Figures 8 and 9. This function is predicted if the rise-time constant and amplitude of the ADP response are such that a new input spike does not lead to a reactivation spike within the same theta cycle. If neurons exist that respond in this way, then a reverse-order persistent spiking buffer provides a mechanism for the repeated presentation of item spikes needed to establish a backward association with spike-timing–dependent plasticity. When learning from realistic training input in which the stimuli of one trial are presented once in a specific order, such backward associations are needed in models of goal-directed behavior that rely on converging forward and reverse spread of activity through neurons, each of which represent an item in a sequence of encountered items (such as places or stimuli) (Hasselmo 2005; Koene and Hasselmo 2005), or in models of context-dependent retrieval of episodes (Hasselmo and Eichenbaum 2005).
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The Effect of the Theta Frequency and of the Strength of Competitive Inhibition on Buffer Capacity
In Figure 10, we show that the capacity of the buffer is increased to 7 items when we lower the frequency of the theta rhythm that septal inputs are assumed to provide to 5 Hz. By retaining the same strength of the recurrent inhibition in the network, more gamma cycles were nested within the depolarizing reactivation phase interval of the extended theta cycle duration.
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When instead we manipulate only the conductance parameter for the inhibitory response produced at pyramidal cells due to recurrent input from the gamma interneuron network, we show that strengthening that inhibition leads to a smaller capacity of STM (Fig. 11). Stronger gamma inhibition delays the spiking of successive spike patterns maintained in the buffer, which results in a lower observed gamma frequency. Fewer gamma cycles at that frequency fit into the depolarizing reactivation phase interval of the theta rhythm.
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Conversely, the capacity of STM may be increased if gamma inhibition is weakened. Given a specific theta frequency, the capacity of STM that can be achieved in this way is limited by a critical value for the strength of recurrent gamma inhibition. Below this value, the competitive inhibition that one pattern of pyramidal spikes elicits by activating the gamma interneuron network is insufficient to maintain separation from the following pattern of buffered spikes by a minimum time interval. In Figure 12, we show that STM function deteriorates with weak gamma inhibition (Ggamma < 2.5 ns). After few theta cycles, initially distinct patterns of spikes merge to spike simultaneously. For reliable buffer performance, a greater minimum strength of gamma inhibition is needed when we take into account the influence of noise in biological systems, such as in several of the simulations described in following results.
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The Effect of Interference between Item Representations
We considered the case where the sets of spikes that are consecutively elicited by a sequence of afferent input are not completely distinct. A pyramidal cell of the buffer that has spiked and experiences ADP may spike due to afferent input. When that happens, the ADP response is reset in order to model the resetting of internal calcium concentration of the cell as a consequence of the spiking action. The afferent input therefore determines the new phase of the buffer cycle at which a spike is reactivated, and this spike no longer contributes to the same representation of a buffered item. This is shown for the case of 2 items with overlapping spiking representations, C and F in Figure 13. When the spiking representation of F entered the buffer, the representation of C was reduced to a smaller subset of spikes (C'). If C is a known item that has been previously encoded in a recurrent network, then the reduction of C may be countered by autoassociative retrieval into the buffer (Koene 2006b, in preparation). The spike representation and reactivation of the new item F were not affected.
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If afferent input causes spikes that match the complete representation of a buffered item, then that item will reactivate only in the temporal order (buffer position) of the new input. The original temporal position of the item representation that reappeared while buffered is lost and the sequence shifts to fill the blank location. If the same input is presented multiple times consecutively, then buffer content will not appear to change and will contain only one instance of the corresponding item representation. No effect is noticeable if novel input presents the same set of spikes that represent the first item in a buffer that is filled to capacity.
This loss of a previous instance of an item representation that matches new input was simulated in prior work for a delayed nonmatch to sample task (McGaughy et al. 2005) to show that it may explain specific errors in the observed behavior of rats (McGaughy et al. 2004). In the experiment, rats were trained to sniff consecutively at 2 different odors (A and B). Following a delay, the rats were given a test odor, after which the task required that the rat dig in one of the original 2 containers to point out the one with the nonmatching odor. Rats with prefrontal cortex lesions, but intact EC, displayed a peculiar performance error. If trained rats were presented with odors A and B, then with a test odor matching A, they had no difficulty identifying the separate test container with odor B. When these rats were given a test odor matching the second of the 2 sample odors (B), then they were confused by the repetition and were unable to perform the task. They eventually picked a container at random (chance-level performance).
We propose that the observed inability of lesioned rats to perform the nonmatch to sample task may be explained by reliance on a buffer with 2 specific characteristics: 1) Only one instance of a specific pattern of spiking neurons can be maintained as part of a buffered sequence. 2) Presenting a novel pattern of spikes may cause the displacement and therefore the forgetting of a previously buffered item representation. The effect of these 2 functional constraints is demonstrated most clearly in a simplified simulation where the sequence buffer of a virtual rat has a capacity of only 2 items.
For virtual rats with the simplified 2-item buffer, our model predicts that a reduction of buffer content is caused by the presentation of the same stimuli that elicited maintenance of that last (i.e., second) item in the buffer. As explained above, this is predicted for all cases where spiking due to new input matches a spiking representation that is maintained in the buffer. Each buffered item may represent knowledge of an odor and an associated container. When the test phase involves input of B, then the replacement mechanism removes A from the full buffer. Only one item (B) remains in the buffer. According to our model, the rats with prefrontal lesions are left with insufficient information in that case to perform the 2 comparisons that are expected in the delayed non-match to sample task.
We do not wish to imply that the full capacity of the short-term buffer in rats is 2 items, so we follow the 2-item example with a simulation where the capacity is a more plausible 3 items. In that case, we presume that the third buffered item represents some interceding event, such as a perception of the switch from sample presentation to the delay and performance parts of the task. We show the results of simulation with both capacities, 2-item capacity in Figure 14a and 3-item capacity in Figure 14b. Simulating a buffer limited to 2 items is done with a phase offset of 53 ms for the transmission modulation of buffer output into neurons detecting the full buffer state, and the conductace amplitude of the input from these detector neurons to the replacement interneurons is raised to G = 1.0 ns.
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A Reliable Model of STM Deals Gracefully with Noise
We evaluated the effect of noise on the function of the STM buffer when the reactivation of novel patterns of simultaneous spikes relies entirely on persistent firing without synapse-dependent retrieval. Noise was added through simulated current clamps of individual neurons driven by a first-order autoregressive process (a model for the response to noise that is similar to a random walk) with Poisson distribution, a mean value of 0, amplitude 1 pA, and regression parameter 0.5. With this noise, we evaluated the statistical probability of errors in the buffered patterns of spikes by performing 50 simulation runs, each over 5000 ms, presented with 6 stimuli that were represented by patterns consisting of between 2 and 8 simultaneous spikes with a total of 28 active neurons. Of the 50 runs, 27 simulation runs produced no errors. The mean error quantified as all missing or extra spikes at the end of a simulation run was 1.22 spikes with a standard deviation of 1.8 (a bit error rate of 0.044). The first item pattern presented to the buffer contained errors in 15 of the 23 runs that exhibited errors due to noise. In only 2 of the simulation runs did more than one spike pattern contain errors. In both of these runs, the number of patterns involved was 2, and in both cases, the first item pattern presented to the buffer was one of these. Never was an entire spike pattern lost to noise.
During some simulation runs, individual spikes of the first pattern in the STM buffer stopped reactivating if noise delayed reactivation sufficiently so that the competitive recurrent inhibition caused by the remaining spikes suppressed reactivation of the delayed spike until its ADP began to fall. In effect, the spike was removed from the buffer in the same manner as items are removed by the replacement mechanism (Fig. 15). A similar error appears occasionally in items other than the first if reactivation in the same theta cycle as the presentation of afferent input fails.
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When noise produces a delay in the reactivation of individual spikes, delayed spikes may end up in a separate gamma cycle, effectively adding another item in the buffer that is then sustained in that order (Fig. 16). For novel items, encoding with such noise effects may separate a single item representation (normally stored through STDP by the strengthening of autoassociative synaptic connections) into multiple parts that are subsequently stored as an episode.
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In general, the first item in the STM buffer is more susceptible to noise-induced errors than the following items that are flanked by gamma inhibition and for which reactivation is driven by combinations of depolarization by theta modulation and ADP that significantly exceed the threshold potential. The membrane potentials of these items exceed the threshold potential before reactivation of spiking, as their spiking is actively delayed by recurrent inhibition. It is notable that even in the presence of relatively strong noise (with values up to ±10 pA), several of the spikes in each item pattern tend to survive the difficult insertion into the buffer and continue to reactivate in the correct order (Fig. 17a). As even stronger noise is added (values that reach ±60 to ±70 pA), the duration of item maintenance in the STM buffer is limited by the dropout of spikes (Fig. 17b).
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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We hypothesize that persistent firing characteristics of pyramidal cells in ECII can provide a reliable short-term buffer for tasks in which encoding of autoassociative and episodic memory depends on the rapid acquisition and maintenance of novel item representations. We propose an explicit mechanism for the FIFO ordered replacement of items in a buffer filled to capacity, and we show that the buffer mechanism can be specified with a range of parameter values and modifications of its sp