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Exhibit 31.1 Certification Pursuant to Rule 13a-14 of the Securities Exchange Act of 1934,
as adopted pursuant to Section 302 of the Sarbanes-Oxley Act of 2002 I, Christopher G. Chavez, certify that: 1. I have reviewed this quarterly report on Form 10-Q of Advanced Neuromodulation Systems, Inc.; 2. Based on my knowledge, this report does not contain any untrue statement of a material fact or
omit to state a material fact necessary to make the statements made, in light of the circumstances
under which such statements were made, not misleading with respect to the period covered by this
report; 3. Based on my knowledge, the financial statements, and other financial information included in
this report, fairly present in all material respects the financial condition, results of operations
and cash flows of the registrant as of, and for, the periods presented in this report; 4. The registrants other certifying officer and I are responsible for establishing and maintaining
disclosure controls and procedures (as defined in Exchange Act Rules 13a-15(e) and 15d-15(e)) and
internal control over financial reporting (as defined in Exchange Act Rules 13a-15(f) and
15d-15(f)) for the registrant and have: a) designed such disclosure controls and procedures, or caused such disclosure controls and
procedures to be designed under our supervision, to ensure that material information relating to
the registrant, including its consolidated subsidiaries, is made known to us by others within those
entities, particularly during the period in which this report is being prepared; b) designed such internal control over financial reporting, or caused such internal control over
financial reporting to be designed under our supervision, to provide reasonable assurance regarding
the reliability of financial reporting and the preparation of financial statements for external
purposes in accordance with generally accepted accounting principles; c) evaluated the effectiveness of the registrants disclosure controls and procedures and presented
in this report our conclusions about the effectiveness of the disclosure controls and procedures,
as of the end of the period covered by this report based on such evaluation; d) disclosed in this report any change in the registrants internal control over financial
reporting that occurred during the registrants most recent fiscal quarter (the registrants fourth
fiscal quarter in the case of an annual report) that has materially affected, or is reasonably
likely to materially affect, the registrants internal control over financial reporting; and 5. The registrants other certifying officer and I have disclosed, based on our most recent
evaluation of internal control over financial reporting, to the registrants auditors and the audit
committee of registrants board of directors (or persons performing the equivalent function): a) all significant deficiencies and material weaknesses in the design or operation of internal
control over financial reporting which are reasonably likely to adversely affect the registrants
ability to record, process, summarize and report financial information; and b) any fraud, whether or not material, that involves management or other employees who have a
significant role in the registrants internal control over financial reporting. Date: May 9, 2005 /s/ Christopher G. ChavezName: Christopher G. ChavezTitle: Chief Executive Officer