Journal of Robotics, Networking and Artificial Life, Vol. 1, No. 4 (March 2015), 312-315

Associative Memory with Class I and II Izhikevich ModelYoshika Osawa

Department of Electrical Engineering and Information Systems, The University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Takashi Kohno

Institute of Industrial Science, University of Tokyo
4-6-1, Komaba, Meguro-ku, Tokyo, 153-8505, Japan
E-mail: osawa@sat.t.u-tokyo.ac.jp, kohno@sat.t.u-tokyo.ac.jp
www.sat.t.u-tokyo.ac.jp/~kohno/

Abstract
Spiking neural network is a system that qualitatively reproduce the nervous system. It was shown in previous
researches that the performance of associative memory task in all-to-all connected networks is higher when they arecomposed of Class II neurons than Class I neurons. The Izhikevich model in its Class II mode, however, does nothave this performance boost. In this study, we focus on phase resetting curve as an index that reflects neuronalproperties related to neuron classes in detail.
Keywords: Spiking neural network, associative memory, Izhikevich model, Phase resetting curve, Hodgkin's
classification

1. Introduction
Nervous system has robust, autonomous, and powerefficient information processing capacity. Neurons
generate overshoot of their membrane potential called
spike, when sufficiently large stimulus is applied, which
travel through their axon and transmit signals to postsynaptic neurons via synapses by transmitter.
There are several approaches in modeling neuronalactivities. Conductance-based models that focus on thecurrent through the cell membrane such as the HodgkinHuxley model1 can reproduce those activities precisely.
Since this type of models are composed of complex
differential equations, large-scale simulation isimpractical. The Leaky Integrate-and-Fire (LIF) model2
represents the spike by resetting of membrane potential.
This method yields simple and low-dimensional systemequations. It is very easy to simulate using this model
but it reproduces limited aspects of neuronal behaviors.
In the meanwhile simplified expression while
maintaining the mechanism of the conductance-basedmodels develops the qualitative modeling. One of them,
Digital Spiking Silicon Neuron (DSSN) model3 iscomposed of two differential equations that expresses
the spike generation process without reset of systemvariables. The Izhikevich (IZH) model4 captures only
the spike decision process qualitatively, and represents
the spike by reset of its system variables. It is possible
to reproduce a variety of spike patterns.
Neurons are classified by characteristics of the periodic
firing in the Hodgkin’s classification5. Associative
memory simulation using the DSSN model built by Li
et al.6 showed that an all-to-all connected network ofClass II neurons has higher recall ability than that of

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312Fig. 1. The phase plane of Izhikevich model in (a) Class I and(b) Class II modes.
Fig. 3. Change of the shape of the phase resetting curve
depending on the resetting parameter d.

Fig. 2. Phase resetting curve of Izhikevich model in (a) Class Iand (b) Class II modes.

Class I neurons. Higher recall ability here means that an
original stored pattern is recalled from an associatedinput pattern with higher error. We configured similar
associative memory using the IZH model and evaluatedthe difference of its performance by focusing on theshape of the phase resetting curve (PRC)7.

Fig. 4. Time course example of membranetransmitter concentration, and synaptic output.

potential,

2. Neuron Model and Synapse Model
2.1. Izhikevich Model
The IZH model captures the mechanism of spike
decision process based on its bifurcation structure.
Saddle-node and Hopf bifurcations produce Class I andII properties, respectively. In the IZH model, thismechanism is described by two differential equation (1),and membrane potential v is reset when it exceeds 30.

neuron arise periodic firing before he intersection of thetwo lines disappear although no occur in Class I. The
PRC is a plot of spiking phase shift in a neuron atperiodically spiking state induced by sufficiently small
pulse stimulus. Its horizontal axis is the phase at which
the stimulus is applied. Fig. 2 shows the PRC of the IZH
model. There are two types of the shape. The Type 1 isalways positive, and Type 2 is biphasic. The PRC of theIZH model is Type 1 and 2 in Class I and II models,respectively. The PRC of Class II has a jump point
induced by reset. Its jump is reduced by reset parameterd (Fig. 3).

(1)
2.2. Synapse ModelHere, u is a variable that represents membrane recovery
the parameters a, b, c, and d are constants that specify
the model’s firing property, and I represents a stimulus
input. The phase portraits of Class I and II neuron
modes of this model are described in Fig. 1. Class II

Synapses transmit signals to post-synaptic neurons.
Chemical synapses release transmitter when its presynaptic neuron fires. In our model, the neurotransmitteris in the same manner as Li et al., approximated to vary
in a pulse form while the spike exceeds a threshold. The
expression of synaptic output is shown in Eq. (2)8.

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Fig. 5. Stored patterns in our associative memory simulation.

Fig. 6. Examples of input patterns, generated by applying
errors to the stored pattern 1.

(2)
are constants that represent the time
Here, andconstant of rising and phase of synaptic outputs,respectively. The amount of transmitter [T] is set to 1
when v > 0, and 0 when v < 0. Figure 4 represents
example of time variation of the membrane potential,
neurotransmitter concentration, and the synaptic current.

Fig. 7. (a) Waveform example of membrane potential and (b)its raster plot. Input pattern is the upper left in Fig. 6. Blue andred waveforms and squares correspond to the black and white
dots of the stored pattern 1 in Fig. 5.

3. Simulation
We examined the performance of the associative
memory composed of Class II Izhikevich model varying
resseting parameter d. The associative memory iscomposed of all-to-all connected networks and outputs a
stored pattern similar to the input data. In this work, ppatterns of N dots which are orthogonal to each otherare stored (p = 4, N = 256). The patterns are listed inFig. 5. The weightof the synapses from i-th to j-thneuron is calculated by Eq. (3).
(3)
Input patterns are generated by applying random errors
to a stored pattern (Fig. 6). We made 100 input patterns
for every error rate. The error level vary from 5 to 50%
with basically 5% steps and 1% in a region where therecall rate starts to decrease rapidly.
Firstly, we apply a pulse stimulus only to the neurons
corresponding black dot. Then, stimulus current to allneurons was set to the value that induce periodic firing.

Fig. 8. Time course of the overlap
Fig. 7(a).

that corresponds toFigure 7(a) is an example of time series data of 256
neuron’s membrane potential when the upper left
pattern was input. Figure 7(b) shows the raster plot of
this example. In this figure, we can see that at earlyphase of the recall process the neurons fire relatively inan asynchronous manner. The synchronicity increases
as the neurons continue firing. And finally the neurons
that correspond to black (white) dots fire synchronously.
These two synchronously firing neuron groups areantiphasic. To assess this quantitatively, we used
overlapthat calculates the matching degree of uth stored pattern from the phase in the period of eachneuron at a time t (Eq. (4)).

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314

(4)

PRC shown in Fig. 3, the balance between the regionsof positive and negative value of the phase shift may be
important for the performance of associative memory.
4. Conclusion
We have configured the associative memory consistingof 256 neurons of the IZH model. The performance ofClass II network was worse than Class I, but theperformance of the associative memory is improved byincreasing the reset parameter d, by which the shape of
the PRC changes. In the future, the validation by aphase model representing the shape of the quantitative
PRC will be performed to quantitatively evaluate how
the shape of the PRC contributes to the performance of
the associative memory.
Acknowledgements
Fig. 9. Overlap averageof each error level. (a) From 5%
to 50% by 5%. (b) From 30% to 45% by 1%.

This work was supported by JSPS Grant-in-Aid forscientific Exploratory Research Number 25240045
References

Fig. 10. Maximum input error level with which the recall iscorrect (> 0.95 for all trial).

=1Here, is the time that k-th spike is generated.
indicates that the u-th pattern is recalled successfully.
The time course of
calculated by applying Eq. (4)to the data in Fig.7 is shown in Fig. 8. This result
indicates that the stored pattern 1 is recalled correctly.
We performed simulation for 100 input patterns for eacherror level. The reset parameter d of each neuron wasvaried from 0 to 200 by 20 in this simulation. Figure 9shows the overlap averagefor each error level.
Since that the change ofis relatively large when
input error level is 30-45%, we varied the error level by
1% step. We define that the recall is correct when>
0.95 for all trials. Figure 10 plots the maximum input
error rate from which our network recalled the correct
pattern. As shown in Fig. 10, it got almost saturated in
the vicinity of d = 120. Compared with the shape of

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