Memristive model of amoeba learning

Recently, it was shown that the amoebalike cell Physarum polycephalum when exposed to a pattern of periodic environmental changes learns and adapts its behavior in anticipation of the next stimulus to come. Here we show that such behavior can be mapped into the response of a simple electronic circuit consisting of a $LC$ contour and a memory-resistor (a memristor) to a train of voltage pulses that mimic environment changes. We also identify a possible biological origin of the memristive behavior in the cell. These biological memory features are likely to occur in other unicellular as well as multicellular organisms, albeit in different forms. Therefore, the above memristive circuit model, which has learning properties, is useful to better understand the origins of primitive intelligence.

Figure 1

(Color online) View of Physarum during movement. (a) Physarum moves via shuttle streaming, a process of periodic flow of the ectoplasm and endoplasm, with a greater net flow toward the direction of movement or the anterior end. The ectoplasm contains radial and longitudinal actin-myosin fibers whose oscillating contractions help produce the pressure gradient which drives shuttle streaming [7]. (b) During movement, the anterior of Physarum may develop a sheet of gel that inhibits streaming. However, when the pressure gradient builds to a certain threshold, the gel can break down allowing for the formation of new channels of flow. These channels may become new permanent veins for streaming.

Figure 2

(Color online) Electronic circuit that models amoeba’s learning. (a) Schematic representation of the learning circuit made of four fundamental two-terminal circuit elements: resistor $R$, inductor $L$, capacitor $C$, and memristor $M$. (b) Sketch of the selected memristor function $f$ which depends on the voltage applied to the memristor (more details are given in the text). It is defined as $f\left(V\right)=-\beta V+0.5\left(\beta -\alpha \right)\left(\left|V+V_T\right|-\left|V-V_T\right|\right)$, where $\alpha$ and $\beta$ are positive constants and $V_T$ is a threshold voltage.

Figure 3

(Color online) Simulations of the circuit response to applied pulse sequences. (a) Arbitrary pulse sequence. (b) Learning pulse sequence. The calculations were made using the following system parameters: $R=1\Omega$, $L=2\text{H}$, $C=1\text{F}$, $M_1=3\Omega$, $M_2=20\Omega$, $\alpha =0.1\Omega /\left(\text{Vs}\right)$, $\beta =100\Omega /\left(\text{Vs}\right)$, and $V_T=2.5\text{V}$. The applied pulse sequence was selected in the form $V\left(t\right)=V_F-V_p{\sum }_i\left\{\text{cos}\left[2\pi \left(t-t_i\right)/W_p\right]-1\right\}\theta \left(t-t_i\right)\theta \left(t_i+W_p-t\right)/2$, where $V_F$ is the voltage corresponding to the standard (favorable) conditions, $V_p$ is the pulse amplitude, $t_i$ is the time of start, and $W_p$ is the pulse width. In the calculations, we used $V_F=0.1\text{V}$, $V_p=2\text{V}$, and $W_p=5\text{s}$. The $t_i$’s can be identified from the figure.

Figure 4

(Color online) Modeling of the spontaneous in-phase slow-down responses. This plot demonstrates that stronger and longer-lasting responses for both spontaneous in-phase slow down and spontaneous in-phase slow down after one disappearance of the stimulus are observed only when the circuit was previously trained by a periodic sequence of three equally spaced pulses as present in $V_2\left(t\right)$. The applied voltage $V_1\left(t\right)$ is irregular and thus three first pulses do not “train” the circuit to learn. The simulation parameters are as in Fig. 3. Lines were displaced vertically for clarity. Arrows are used to define the correct vertical axis for each line.

Figure 5

(Color online) Resistance of memristor calculated at two different values of the parameter $\beta$ in the function $f\left(V_C\right)$ as a function of time interval $\tau$ between the pulses. Here, we plot the value of the memristance $M$ right after the last (third) pulse in the learning sequence. The memristor state significantly changes when pulse period is close to the $LC$ contour frequency. The simulation parameters are as in Fig. 3.