Information Maximization in a Feedforward
Network Replicates the Stimulus Preferenceof the Medial Geniculate and the Auditory
Cortex
Takuma Tanaka(B)The Center for Data Science Education and Research,
Shiga University, 1-1-1 Banba, Hikone, Shiga 522-8522, Japantanaka.takuma@gmail.com

Abstract. Central auditory neurons exhibit a preference for complex
features, such as frequency modulation and pitch. This study shows thatthe stimulus preference for these features can be replicated by a networkmodel trained to maximize information transmission from input to output. The network contains three layers: input, ﬁrst-output, and secondoutput. The ﬁrst-output-layer neurons exhibit auditory-nerve neuron-like
preferences, and the second-output-layer neurons exhibit a stimulus preference similar to that of cochlear nucleus, medial geniculate, and auditory cortical neurons. The features detected by the second-output-layer
neurons reﬂect the statistical properties of the sounds used as input.
Keywords: Information maximization · Auditory information
processing · Auditory cortex · Pitch selectivity · Frequency modulation
selectivity

1

Introduction

Neurons in the auditory system of the brain can detect multiple dimensionsof the complex features of sounds. For example, human speech is composedof a series of combinations of features such as phoneme, tone, stress, length,
and prosody. The precise timing and temporal variation of these features convey
critical information about the content of speech. To correctly comprehend humanspeech, an auditory system must be able to detect the features precisely andencode these features in the ﬁring patterns of neurons.
The neural encoding of sensory inputs has been intensively studied bothexperimentally and theoretically. Experiments on auditory information processing have revealed that auditory-nerve neurons respond to sine wave-like tones[1] and that central auditory neurons encode complex features such as soundintensity and pitch. More than half of medial geniculate neurons in cats exhibit
non-monotonic rate–sound-intensity functions [2], which implies that these neurons can be interpreted as intensity-coding neurons. Bendor and Wang reportedc Springer International Publishing AG 2016

A. Hirose et al. (Eds.): ICONIP 2016, Part IV, LNCS 9950, pp. 183–190, 2016.
DOI: 10.1007/978-3-319-46681-1 22

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T. Tanakathat neurons in the auditory cortex of marmoset monkeys responded to bothpure tones and missing fundamental harmonic complex sounds and called them
pitch-selective neurons [3]. This type of high-level feature selectivity is thought
to have emerged as a result of the integration of simpler low-level feature selectivity, such as that exhibited by auditory-nerve neurons, in a circuit with hierarchical structure. Hierarchical structure has been widely used in the theoreticalmodeling of visual information processing [4–7]. Theoretical studies of visual
processing have demonstrated that stimulus selectivity to complex features suchas the boundary between two gratings emerges in a generative model of naturalscenes [8,9] and in network models that maximize the amount of information
conveyed by the output [7,10]. These studies suggest that feature representationin the auditory system can be understood on the basis of a similar framework.
In fact, a previous model of auditory feature representation [11] showed that
units representing complex auditory features emerged in a generative model ofsounds. However, the “spikes” in this model were generated by the maximum
likelihood estimation, and therefore whether neurons can perform such a computation remains unclear. Moreover, complex auditory feature detection has notbeen treated in terms of maximizing information transmission.
Therefore, this study examines auditory feature detection in a feedforward
network of rate-coding neuron models using an algorithm based on the information maximization principle. The network consists of three layers: input, ﬁrstoutput, and second-output. Short waveforms from a natural sounds dataset anda human speech corpus are provided to the input layer. The ﬁrst-output-layer
neurons respond to the wavelet-like waveforms. The second-output-layer neurons encode more complex features, such as pitch, tone intensity, and upwardand downward frequency modulation. The selectivity of these model neurons forthese complex features is comparable to that of experimentally reported cochlear
nucleus, medial geniculate, and auditory cortical neurons. These results suggestthat the central auditory neurons can be understood in terms of information
maximization and that an extended network model based on the information
maximization principle could replicate the more complex feature detection of
the auditory cortices.

2

Model

The Pittsburgh natural sounds dataset [12] was used as the natural-sound input,
and the Priority Areas “Spoken Dialogue” Simulated Spoken Dialogue Corpus(PASD) was used as the human-speech input. The former was down-sampled to
11 kHz, and the latter was down-sampled to 8 kHz. The input time series wasshifted and scaled to have zero mean and unit variance. Consecutive samples(N = 200) were randomly chosen from the input time series to be used as input
to the network at each time step. The network model and learning algorithm
described previously [7] were used. The network consists of input, ﬁrst-output,
and second-output layers, each containing N model neurons. The value of thei-th input at time step t is xi (t). The states of the neurons in the ﬁrst- andAuditory Stimulus Preference Replicated by Infomax Principle

second-output layers at time t are determined by⎛⎞Nui (t) = f ⎝Vij xj (t)⎠ ,

185

(1)

j=1

⎛
⎞
N

zi (t) = f ⎝Wij (|uj (t)| − |uj |)⎠ ,

and

(2)

j=1

respectively, where Vij is the connection weight from the input neuron j to theﬁrst-output-layer neuron i, Wij is the connection weight from the ﬁrst-outputlayer neuron j to the second-output-layer neuron i, |uj | is the average of |uj (t)|over time, andx(3)f (x) = 2 arctan tanh2is the activation function. The integration time constant τ was set to 104 steps inall simulations. The weights Vij were updated once every 1000 steps. The updatesof the matrices V and W were performed 104 times with  = 0 , 1.9 × 105 times
with  = 100 , and 8 × 105 times with  = 0 using the Newton method describedin [7], where 0 = 10−6 for natural-sound input and 0 = 10−5 for human-speech
input.

3

Results

3.1

First-Output-Layer Neurons

The input and ﬁrst-output layer can be regarded as a network performing independent component analysis [6]. Similar to a previous study of independent component analysis [13], the ﬁrst-output-layer neurons exhibit selectivity to Gaborwavelet-like sound waveforms. Figure 1A shows the column vectors of V−1 , soundwaveforms that the ﬁrst-output-layer neurons are selective for, of the networkin which natural sounds are used as the input. They have Gabor function-likeshapes with diﬀerent frequencies, amplitudes, phases, dispersions, and center

B

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Fig. 1. Preferred sound waveform of 20 ﬁrst-output-layer neurons in networks with (A)natural-sound input and (B) human-speech input.

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T. Tanaka

positions. Figure 1B shows the selectivity of the ﬁrst-output-layer neurons withhuman-speech input. The waves in Fig. 1B resemble sine waves without amplitude modulation because the majority of human speech is composed of vowels.
Selectivity to these sine waves with and without amplitude modulation is similar
to the stimulus preference of auditory-nerve neurons [1,13].
The response of these neurons to continuous sine waves is shown in Fig. 2.
The horizontal and vertical axes represent the frequency and amplitude of sine
waves, respectively. The density represents the maximal value of the ﬁrst-outputlayer neuron output obtained by varying the phases of the sine waves. Almost
all neurons respond to a small-amplitude tone with the preferred frequency andto a wide range of tones if the amplitude is increased. This ﬁgure shows thatthe wavelet-like connection weights result in unimodal, V-shaped tuning curves
with only one preferred frequency [14].

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3.2

Second-Output-Layer Neurons

Figure 3 shows the connection weights from the ﬁrst-output-layer neurons to thesecond-output-layer neurons in the networks with natural-sound and humanspeech inputs. Each box corresponds to a second-output-layer neuron, and eachline in the box corresponds to the connection weight from a ﬁrst-output-layer
neuron to the second-output-layer neuron. The vertical position of a line in thebox represents the preferred frequency of the ﬁrst-output-layer neuron obtainedby ﬁtting the row vector of V−1 with the Gabor function, that is, ω ofg(t) = exp[−(t − t0 )2 /(2πσ 2 )] sin(ωt + φ).

(4)

The horizontal position and the length of the line represent t0 and σ, respectively.
The color represents the value of the connection weight Wij , with red and blue

Auditory Stimulus Preference Replicated by Infomax Principle

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Fig. 3. Connection weights from ﬁrst- to second-output-layer neurons for the networks
with (A) natural-sound input and (B) human-speech input. The second-output-layer
neurons are represented as boxes. Connections from the ﬁrst-output-layer neurons correspond to the horizontal lines, whose vertical position, horizontal position, and length
represent ω, t0 , and σ of the ﬁtted Gabor function, respectively. Red and blue indicate
positive and negative connection weights, respectively. (Color ﬁgure online)

indicating positive (excitatory) and negative (inhibitory) connection weights,
respectively.
The stimulus selectivities exhibited by the ﬁrst-output-layer neurons in the
networks with natural-sound and human-speech inputs are substantially diﬀerent
from each other. Because the diﬀerence in the stimulus preference of the ﬁrstoutput-layer neurons aﬀects the stimulus preference of the second-output-layer
neurons, Figs. 3A and B show completely diﬀerent types of stimulus preferencesin these two networks. Therefore, the results of the two networks with diﬀerent
inputs are presented separately.
Natural Sounds
Frequency-Modulation Selectivity. Second-output-layer neuron a in Fig. 3A
receives positive connection weights from ﬁrst-output-layer neurons selective forlow-frequency tones in the earlier half and for high-frequency tones in the latter half, while it receives negative connection weights from neurons selective forhigh-frequency tones in the earlier half and for low-frequency tones in the latter
half. Therefore, this neuron is selective for a frequency change from a low tone
to a high tone. Selectivity to frequency modulation is found in more than halfof the neurons in the cochlear nucleus [15]. Neuron a appears to correspond tothese cochlear nucleus neurons responsive to frequency modulation.
Intensity Tuning. Figure 4A shows the tuning curves of the second-output-layerneurons trained with natural-sound input. These tuning curves are much morevariable than those of the ﬁrst-output-layer neurons (Fig. 2A). Tuning curve b

T. Tanaka

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in Fig. 4A has two preferred frequencies; this suggests that this neuron receives
strong positive connection weights from two ﬁrst-output-layer neurons with different frequency preferences, which is evident in Fig. 3A (b). This is similar toauditory cortical neurons classiﬁed as double U-shaped tuning curves [16]. The
tuning curve c (Fig. 3A) is similar to circumscribed neurons in the auditory
cortex [16], which do not respond to increased amplitude of sine waves at any
frequency. Therefore, this neuron is selective for a tone with a speciﬁc frequency
and intensity level. Physiologically, this type of neuron is reported to compose
approximately 20 % of the neurons in a cat’s auditory cortex [16].
Human Speech
Pitch Selectivity. Figure 3B shows the connections from ﬁrst- to second-outputlayer neurons in the network with human-speech input. Some of these neurons
receive positive inputs from ﬁrst-output-layer neurons with preferred frequencies that are multiples of a value (neuron d ). That is, these neurons respond to
the pitch of a tone, and, consequently, have intensive multimodal tuning curves(Fig. 4, neuron d ). This type of stimulus preference has been reported in auditory cortical neurons [16]. Because the present learning algorithm maximizes theinformation transmission from input to output, this result suggests that pitch
selectivity in the auditory cortex emerges to encode human and other animal
voices eﬃciently. Indeed, human voices are primarily composed of tones with aprincipal frequency and its higher harmonics. Owing to this statistical propertyof human voices, second-output-layer neurons acquire selectivity to tones withfrequencies f, 2f, 3f, . . . during training.

4

Discussion

This study presented the properties of model neurons in networks trained to maximize the amount of information transmitted to the output layers, using natural

Auditory Stimulus Preference Replicated by Infomax Principle

189

sounds and human speech as inputs. The ﬁrst-output-layer neurons respond towavelet-like stimuli and have unimodal tuning curves. This property is consistent
with the experimentally reported properties of auditory-nerve neurons [1]. It isalso consistent with previous theoretical studies [11,13], showing that waveletlike functions are information-eﬃcient in encoding natural sounds. The properties of the second-output-layer neurons are aﬀected by the type of input. Naturalsounds, which contain abundant abrupt changes in pitch, favor the emergenceof second-output-layer neurons selective for pitch change. In contrast, humanvoices are dominated by continuous waves with higher harmonics, making pitch
selectivity advantageous in encoding information. These properties are also consistent with previous experimental and theoretical results [3,11,15].
The training algorithm of the present model is based on the information
maximization principle, that is, the information conveyed by the output neurons is maximized during training. The fact that neurons in the two output
layers exhibit stimulus preferences similar to the cochlear nucleus, medial geniculate, and auditory cortical neurons suggests that the neurons in the central nervous system have evolved to encode as much information as possible by forming
an information-eﬃcient circuit. This is corroborated by previous studies whichshowed that the properties of simple and complex cells in the primary visualcortex can be replicated by the information maximization model [7,17]. If, assuggested by the present model, the sensory information processing in the central nervous system can be understood in terms of information maximization, amodel with a larger number of layers would replicate and predict the propertiesof neurons in the higher sensory cortices.
Acknowledgments. This work was supported by JSPS KAKENHI Grant Numbers
15H04266 and 16K16123.

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