Journal of Computational Neuroscience

, Volume 40, Issue 2, pp 193–206 | Cite as

A minimum-error, energy-constrained neural code is an instantaneous-rate code

  • Erik C. Johnson
  • Douglas L. Jones
  • Rama Ratnam


Sensory neurons code information about stimuli in their sequence of action potentials (spikes). Intuitively, the spikes should represent stimuli with high fidelity. However, generating and propagating spikes is a metabolically expensive process. It is therefore likely that neural codes have been selected to balance energy expenditure against encoding error. Our recently proposed optimal, energy-constrained neural coder (Jones et al. Frontiers in Computational Neuroscience, 9, 61 2015) postulates that neurons time spikes to minimize the trade-off between stimulus reconstruction error and expended energy by adjusting the spike threshold using a simple dynamic threshold. Here, we show that this proposed coding scheme is related to existing coding schemes, such as rate and temporal codes. We derive an instantaneous rate coder and show that the spike-rate depends on the signal and its derivative. In the limit of high spike rates the spike train maximizes fidelity given an energy constraint (average spike-rate), and the predicted interspike intervals are identical to those generated by our existing optimal coding neuron. The instantaneous rate coder is shown to closely match the spike-rates recorded from P-type primary afferents in weakly electric fish. In particular, the coder is a predictor of the peristimulus time histogram (PSTH). When tested against in vitro cortical pyramidal neuron recordings, the instantaneous spike-rate approximates DC step inputs, matching both the average spike-rate and the time-to-first-spike (a simple temporal code). Overall, the instantaneous rate coder relates optimal, energy-constrained encoding to the concepts of rate-coding and temporal-coding, suggesting a possible unifying principle of neural encoding of sensory signals.


Rate coding Temporal coding Instantaneous rate Sensory coding Energy efficient coding 



This research was supported by National Science Foundation grants EFRI-BSBA-0938007 and IGERT 0903622, research funds from the College of Engineering, UIUC, Coordinated Science Laboratory, UIUC and the Advanced Digital Sciences Center, Illinois at Singapore. Electric fish data were collected in the laboratory of Mark E. Nelson, UIUC, through the National Institute of Health grant R01MH49242 and National Science Foundation grant IBN-0078206. We gratefully acknowledge the availability of rat cortical pyramidal neuron data in the public domain through the International Neuroinformatics Coordinating Facility.

Compliance with Ethical Standards

All applicable international, national, and/or institutional guidelines for the care and use of animals were followed.

Conflict of interests

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Erik C. Johnson
    • 1
    • 2
    • 3
  • Douglas L. Jones
    • 1
    • 2
    • 3
    • 4
    • 5
  • Rama Ratnam
    • 2
    • 3
    • 4
  1. 1.Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Beckman Institute for Advanced Science and TechnologyUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  4. 4.Advanced Digital Sciences Center, Illinois at Singapore Pte. LtdSingaporeSingapore
  5. 5.Neuroscience ProgramUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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