1 Introduction

Memory, as a higher cognitive function of the brain, has become an important research topic in neural information processing. Three kinds of widely-studied memory are long-term memory, short-term memory, and working memory. Cowan^[1] proposed that working memory contained short-term memory, and working memory and short-term memory had a similar neural pro- cessing mechanism. Short-term memory has been maintained to derive from a temporarily activated subset of information in long-term memory, and may decay over time if the acti- vated subset is not refreshed^{[2, 3]}. In the field of neural biology, the long-term potential (LTP) is the best candidate to study the cell mechanism^[4], because the LTP has much in common with long-term memory^[5]. The stimuli with different properties possess different functions^{[6, 7]}. In a large number of experiments, the θ-burst stimulation (TBS) is used as an LTP induction protocol^{[8, 9, 10]}. The TBS is a repeating pattern of short burst of pulses with brief pauses between the bursts^[11]. Previous researches have shown that the magnitude of the LTP can be affected by some factors like frequency, number of pulses, and stimulus intensity. In this work, the TBS is adopted to induce long-term memory on the basis of working memory. Specifically, based on the Camperi-Wang (C-W) model with the Ca²⁺ subsystem-induced bistability^[12], we change the initial stimulus current into the TBS, and assess its impact on the memory mechanism of the model by the control variate method. The researches help to reveal how to form long-term memory with the TBS on the basis of working memory.

2 Model and method

We adopt the C-W model with the Ca²⁺ subsystem-induced bistability^[12] and the improve- ments of the firing-rate network proposed by Liang et al.^[13]. The network consists of N neurons, which are labeled by their preferred cue locations or memory field θ_i in the range from -π to π. In this limit, the equation of the firing rate r(θ_i, t) is

where τ_r is the integration time. The functions f(r) and g(I) stand for the intrinsic properties of the neuron. By setting $\frac{dr}{dt}=0$, the steady state of this system can be obtained, and thereby the neuronal input-output relation is r = g(I). The piece-wise linear function form of g(I) is

The total input to each neuron I consists of an external input I_ext for the transient cue and a recurrent synaptic input I_syn. Since the Ca²⁺ release might increase the efficacy of the synaptic input, the input I can be represented as follows:

where N_Ca represents the intracellular Ca²⁺ concentration.

The external input I_ext is

where I₀ is the bias input, I_cue is the cue stimulus, θ₀ represents the cue position, and p is the stimulus shape exponent. Here, we define I_cue as the forms of the TBS, which is simulated by the cycle of the square signal. The function “square” in MATLAB is used to make the square signal with the specific frequency, amplitude, and duty ration. The initial TBS is as follows: the frequency of the square signal is set to be 200Hz, the amplitude is from 0 to 1, the duty ration is 20%, the sustain is 0.05 s with the starting time at 1 s (consisting of 10 pulses), the interval period is between 1.05 s and 1.2 s (the signal value is 0 during the interval period), the activity of 1 s to 1.2 s is defined as a cycle, and five cycles (consisting of 50 pulses) are repeated. The synaptic input I_syn is given by the convolution over the weights and firing rates as follows:

The static connectivity weight W between two neurons is given by

where W_I and W_E are the inhibitory strength and the excitatory strength, respectively, and q is the weight shape exponent. △W is the amendment of the synaptic weights defined by^{[14, 15]} where τ_w is the dynamic synaptic time, and k is the dynamic synaptic proportional coefficient. The function of the Ca²⁺ subsystem mainly depends on the level of the second messenger inositol 1, 4, 5 trisphosphate (IP3). A balance equation about the evolution of Ca includes the release of the Ca²⁺ concentration through the IP3 sensitive Ca²⁺ channel (IP3R) into the cytosol (J_IP3R), the removal of the Ca²⁺ concentration (J_SERPM) induced by the reuptake of the SERCA pumps from the cytosol, a leak flux concentration from the endoplasmic reticulum (ER) to the cytosol (J_Leak), and an important contribution to the Ca²⁺ influx concentration from the network activity (J_syn)^[16], i.e., where In the above equations, N_IP3 is the IP3 concentration, h represents the proportion of the IP3R which is not inactivated by the Ca concentration, V_IP3R is the IP3R Ca²⁺ flux, k_SEPRM is the Ca²⁺ pump sensitivity, V_Leak is the Ca²⁺ leak flux, τ_h is the IPR3 inact time, N_CaER is the ER Ca²⁺ concentration, k_inh is the IP3R Ca²⁺ inhibition constant, k_act is the IPR3 Ca²⁺ activation constant, m_∞ is the the fast activation, and α is the coupling parameter. The number of the neurons is 128. In this paper, we use MATLAB (R2011b) with a time step of 0.001 s. The bistability of the Ca²⁺ subsystem can be guaranteed when the level of the second messenger IP3 is chosen from the interval between 0.49 and 1.28. The parameter N_IP3 is taken to be 0.562 to ensure that the Ca²⁺ subsystem is bistable. Other parameters are taken as follows: If the information stored in short-term memory is further activated, long-term memory may be formed^[17]. Cowan believed that working memory contained short-term memory and both working memory and short-term memory had a similar neural processing mechanism^[1]. Based on this, it is considered that the information held in working memory can be transformed into long-term memory after rehearsal, coding, and linking to individual experiences. We hope to transform working memory into long-term memory in the model by adjusting the comprehensive effect of the stimulus properties.

3 Results

The initial form of the TBS (I_cue) studied in this paper is shown in Fig. 1, where t is the time. It consists of ten pulses at 200Hz repeated with 200ms inter-burst-intervals, where the cycle is 5, the amplitude is 1, and the duty ration is 50%. Some parameters remain unchanged, including the stimulus frequency 200Hz, the pulse number of each cycle 10, and the cycle interval 200ms. However, the cycle, amplitude, and duty ration are adjusted according to the control variate method so as to evaluate the impact of these three stimuli properties on the memory model. The starting time is changed to 0 s in the following study.

First, set the TBS with the amplitude 1.8 and the duty ration 50%. Comparing the dif- ferences between the network activities under the circumstances of the stimulus cycle 10 (see Fig. 2(a)) and 30 (see Fig. 2 (c)) reveals the impact of the stimulus cycle on the memory per- formance. It is shown that with the increase in the stimulus cycle, the maximum firing rate of the network activities grows and the memory maintaining time delays. From the projections of Figs. 2(a) and 2(c) in the tr-plane (see Figs. 2(b) and 2(d)), we can see that the maximum of the neuron firing rate increases from 6 to 10 and the period of the network activities with the sustained high firing patterns extends 6 s.

Then, set the TBS with the cycle 30 and the duty ration 50%. From the differences between the network activities under the circumstances of the stimulus amplitude 1.6 (see Fig. 3(a)) and 1.8 (see Fig. 3(c)), we can see that the effects of the stimulus amplitude on the memory performance can be revealed. From Figs. 3(b) and 3(d), we can see that the maximum of the neuron firing rate increases from 9 to 10 and the period of the network activities with the sustained high firing patterns extends from 21 s to 22 s.

At last, set the TBS with the cycle 30 and the amplitude 1.4. From the differences between the network activities under the circumstances of the stimulus duty ration 65% (see Fig. 4(a)) and 80% (see Fig. 4(c)), we can see that the effects of the stimulus amplitude on the memory performance can be revealed. From Fig. 4(d), we can see that the peak of the neuron firing rate exceeds 12 and sustains a high firing rate for 2 s after reaching the peak and the memory time exceeds 4 s (see Figs. 4(b) and 4(d)).

The study shows the adjustment of the comprehensive effect of three kinds of stimuli prop- erties, i.e., cycle, amplitude, and duty ration, of the TBS and that working memory can be transformed to long-term memory. First, set the stimulus properties as the cycle 30, the am- plitude 1.8, and the duty ration 50%, and get the network firing pattern (see Fig. 5(a)) and its corresponding projection in the tr-plane (see Fig. 5(b)). It can be seen that the memory time is 21 s, belonging to working memory. Then, set the stimulus properties as the cycle 30, the amplitude 1.9, and the duty ration 50%, and get Figs. 5(c) and 5(d). It is shown that the memory time is 70 s, meeting the requirement of long-term memory. That is to say, when the stimulus is set as the cycle 30 and the duty ration 50%, the amplitude 1.9 is the critical value to produce long-term memory.

4 Conclusions

The specific stimuli is used in a working memory model to produce long-term memory. The improved C-W model with the Ca²⁺ subsystem-induced bistability is adopted, and the changes of the memory mode in this model are studied under the adjustment of the properties such as cycle, amplitude, and duty ration of the TBS. The research shows that changing the network firing pattern has an effect to some extent on the memory maintaining time. Increasing the cycle, amplitude, and duty ration will increase the firing rate of the network activities and make the memory last longer. When the TBS is set as cycle 30 and duty ration 50%, and the amplitude increases to 1.9, the working memory model can be activated to produce long-term memory. The mechanism of memory formation can be further studied on the basis of our theoretical analysis.